:: RINGCAT1 semantic presentation
:: deftheorem Def1 RINGCAT1:def 1 :
canceled;
:: deftheorem Def2 defines linear RINGCAT1:def 2 :
theorem Th1: :: RINGCAT1:1
canceled;
theorem Th2: :: RINGCAT1:2
canceled;
theorem Th3: :: RINGCAT1:3
:: deftheorem Def3 defines dom RINGCAT1:def 3 :
:: deftheorem Def4 defines cod RINGCAT1:def 4 :
:: deftheorem Def5 defines fun RINGCAT1:def 5 :
Lemma3:
for b1 being non empty doubleLoopStr holds id b1 is linear
:: deftheorem Def6 defines RingMorphism-like RINGCAT1:def 6 :
:: deftheorem Def7 defines ID RINGCAT1:def 7 :
:: deftheorem Def8 defines <= RINGCAT1:def 8 :
Lemma6:
for b1 being RingMorphism holds RingMorphismStr(# the Dom of b1,the Cod of b1,the Fun of b1 #) is RingMorphism-like
:: deftheorem Def9 defines Morphism RINGCAT1:def 9 :
Lemma8:
for b1 being RingMorphism holds the Fun of b1 is linear
Lemma9:
for b1, b2 being Ring
for b3 being strict RingMorphismStr holds
( dom b3 = b1 & cod b3 = b2 & fun b3 is linear implies b3 is Morphism of b1,b2 )
Lemma10:
for b1, b2 being Ring
for b3 being Function of b1,b2 holds
( b3 is linear implies RingMorphismStr(# b1,b2,b3 #) is Morphism of b1,b2 )
Lemma11:
for b1 being RingMorphism holds
ex b2, b3 being Ring st
( b2 <= b3 & dom b1 = b2 & cod b1 = b3 & RingMorphismStr(# the Dom of b1,the Cod of b1,the Fun of b1 #) is Morphism of b2,b3 )
theorem Th4: :: RINGCAT1:4
canceled;
theorem Th5: :: RINGCAT1:5
theorem Th6: :: RINGCAT1:6
Lemma14:
for b1, b2 being Ring
for b3 being Morphism of b1,b2 holds
not ( b1 <= b2 & ( for b4 being Function of b1,b2 holds
not ( b3 = RingMorphismStr(# b1,b2,b4 #) & b4 is linear ) ) )
Lemma15:
for b1, b2 being Ring
for b3 being Morphism of b1,b2 holds
not ( b1 <= b2 & ( for b4 being Function of b1,b2 holds
not b3 = RingMorphismStr(# b1,b2,b4 #) ) )
theorem Th7: :: RINGCAT1:7
definition
let c
1, c
2 be
RingMorphism;
assume E16:
dom c
1 = cod c
2
;
func c
1 * c
2 -> strict RingMorphism means :
Def10:
:: RINGCAT1:def 10
for b
1, b
2, b
3 being
Ringfor b
4 being
Function of b
2,b
3for b
5 being
Function of b
1,b
2 holds
(
RingMorphismStr(# the
Dom of a
1,the
Cod of a
1,the
Fun of a
1 #)
= RingMorphismStr(# b
2,b
3,b
4 #) &
RingMorphismStr(# the
Dom of a
2,the
Cod of a
2,the
Fun of a
2 #)
= RingMorphismStr(# b
1,b
2,b
5 #) implies a
3 = RingMorphismStr(# b
1,b
3,
(b4 * b5) #) );
existence
ex b1 being strict RingMorphism st
for b2, b3, b4 being Ring
for b5 being Function of b3,b4
for b6 being Function of b2,b3 holds
( RingMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = RingMorphismStr(# b3,b4,b5 #) & RingMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = RingMorphismStr(# b2,b3,b6 #) implies b1 = RingMorphismStr(# b2,b4,(b5 * b6) #) )
uniqueness
for b1, b2 being strict RingMorphism holds
( ( for b3, b4, b5 being Ring
for b6 being Function of b4,b5
for b7 being Function of b3,b4 holds
( RingMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = RingMorphismStr(# b4,b5,b6 #) & RingMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = RingMorphismStr(# b3,b4,b7 #) implies b1 = RingMorphismStr(# b3,b5,(b6 * b7) #) ) ) & ( for b3, b4, b5 being Ring
for b6 being Function of b4,b5
for b7 being Function of b3,b4 holds
( RingMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = RingMorphismStr(# b4,b5,b6 #) & RingMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = RingMorphismStr(# b3,b4,b7 #) implies b2 = RingMorphismStr(# b3,b5,(b6 * b7) #) ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines * RINGCAT1:def 10 :
for b
1, b
2 being
RingMorphism holds
(
dom b
1 = cod b
2 implies for b
3 being
strict RingMorphism holds
( b
3 = b
1 * b
2 iff for b
4, b
5, b
6 being
Ringfor b
7 being
Function of b
5,b
6for b
8 being
Function of b
4,b
5 holds
(
RingMorphismStr(# the
Dom of b
1,the
Cod of b
1,the
Fun of b
1 #)
= RingMorphismStr(# b
5,b
6,b
7 #) &
RingMorphismStr(# the
Dom of b
2,the
Cod of b
2,the
Fun of b
2 #)
= RingMorphismStr(# b
4,b
5,b
8 #) implies b
3 = RingMorphismStr(# b
4,b
6,
(b7 * b8) #) ) ) );
theorem Th8: :: RINGCAT1:8
for b
1, b
2, b
3 being
Ring holds
( b
1 <= b
2 & b
2 <= b
3 implies b
1 <= b
3 )
theorem Th9: :: RINGCAT1:9
:: deftheorem Def11 defines *' RINGCAT1:def 11 :
for b
1, b
2, b
3 being
Ringfor b
4 being
Morphism of b
2,b
3for b
5 being
Morphism of b
1,b
2 holds
( b
1 <= b
2 & b
2 <= b
3 implies b
4 *' b
5 = b
4 * b
5 );
theorem Th10: :: RINGCAT1:10
for b
1, b
2 being
strict RingMorphism holds
not (
dom b
2 = cod b
1 & ( for b
3, b
4, b
5 being
Ringfor b
6 being
Function of b
3,b
4for b
7 being
Function of b
4,b
5 holds
not ( b
1 = RingMorphismStr(# b
3,b
4,b
6 #) & b
2 = RingMorphismStr(# b
4,b
5,b
7 #) & b
2 * b
1 = RingMorphismStr(# b
3,b
5,
(b7 * b6) #) ) ) )
theorem Th11: :: RINGCAT1:11
theorem Th12: :: RINGCAT1:12
for b
1, b
2, b
3, b
4 being
Ringfor b
5 being
Morphism of b
1,b
2for b
6 being
Morphism of b
2,b
3for b
7 being
Morphism of b
3,b
4 holds
( b
1 <= b
2 & b
2 <= b
3 & b
3 <= b
4 implies b
7 * (b6 * b5) = (b7 * b6) * b
5 )
theorem Th13: :: RINGCAT1:13
theorem Th14: :: RINGCAT1:14
:: deftheorem Def12 defines Ring_DOMAIN-like RINGCAT1:def 12 :
:: deftheorem Def13 defines RingMorphism_DOMAIN-like RINGCAT1:def 13 :
theorem Th15: :: RINGCAT1:15
canceled;
theorem Th16: :: RINGCAT1:16
canceled;
theorem Th17: :: RINGCAT1:17
:: deftheorem Def14 defines RingMorphism_DOMAIN RINGCAT1:def 14 :
theorem Th18: :: RINGCAT1:18
theorem Th19: :: RINGCAT1:19
definition
let c
1, c
2 be
Ring;
assume E30:
c
1 <= c
2
;
func Morphs c
1,c
2 -> RingMorphism_DOMAIN of a
1,a
2 means :
Def15:
:: RINGCAT1:def 15
for b
1 being
set holds
( b
1 in a
3 iff b
1 is
Morphism of a
1,a
2 );
existence
ex b1 being RingMorphism_DOMAIN of c1,c2 st
for b2 being set holds
( b2 in b1 iff b2 is Morphism of c1,c2 )
uniqueness
for b1, b2 being RingMorphism_DOMAIN of c1,c2 holds
( ( for b3 being set holds
( b3 in b1 iff b3 is Morphism of c1,c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is Morphism of c1,c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def15 defines Morphs RINGCAT1:def 15 :
definition
let c
1, c
2 be
set ;
pred GO c
1,c
2 means :
Def16:
:: RINGCAT1:def 16
ex b
1, b
2, b
3, b
4, b
5, b
6 being
set st
( a
1 = [[b1,b2,b3,b4],b5,b6] & ex b
7 being
strict Ring st
( a
2 = b
7 & b
1 = the
carrier of b
7 & b
2 = the
add of b
7 & b
3 = comp b
7 & b
4 = the
Zero of b
7 & b
5 = the
mult of b
7 & b
6 = the
unity of b
7 ) );
end;
:: deftheorem Def16 defines GO RINGCAT1:def 16 :
for b
1, b
2 being
set holds
(
GO b
1,b
2 iff ex b
3, b
4, b
5, b
6, b
7, b
8 being
set st
( b
1 = [[b3,b4,b5,b6],b7,b8] & ex b
9 being
strict Ring st
( b
2 = b
9 & b
3 = the
carrier of b
9 & b
4 = the
add of b
9 & b
5 = comp b
9 & b
6 = the
Zero of b
9 & b
7 = the
mult of b
9 & b
8 = the
unity of b
9 ) ) );
theorem Th20: :: RINGCAT1:20
for b
1, b
2, b
3 being
set holds
(
GO b
1,b
2 &
GO b
1,b
3 implies b
2 = b
3 )
theorem Th21: :: RINGCAT1:21
:: deftheorem Def17 defines RingObjects RINGCAT1:def 17 :
theorem Th22: :: RINGCAT1:22
theorem Th23: :: RINGCAT1:23
:: deftheorem Def18 defines Morphs RINGCAT1:def 18 :
:: deftheorem Def19 defines ID RINGCAT1:def 19 :
definition
let c
1 be
Ring_DOMAIN;
func dom c
1 -> Function of
Morphs a
1,a
1 means :
Def20:
:: RINGCAT1:def 20
for b
1 being
Element of
Morphs a
1 holds a
2 . b
1 = dom b
1;
existence
ex b1 being Function of Morphs c1,c1 st
for b2 being Element of Morphs c1 holds b1 . b2 = dom b2
uniqueness
for b1, b2 being Function of Morphs c1,c1 holds
( ( for b3 being Element of Morphs c1 holds b1 . b3 = dom b3 ) & ( for b3 being Element of Morphs c1 holds b2 . b3 = dom b3 ) implies b1 = b2 )
func cod c
1 -> Function of
Morphs a
1,a
1 means :
Def21:
:: RINGCAT1:def 21
for b
1 being
Element of
Morphs a
1 holds a
2 . b
1 = cod b
1;
existence
ex b1 being Function of Morphs c1,c1 st
for b2 being Element of Morphs c1 holds b1 . b2 = cod b2
uniqueness
for b1, b2 being Function of Morphs c1,c1 holds
( ( for b3 being Element of Morphs c1 holds b1 . b3 = cod b3 ) & ( for b3 being Element of Morphs c1 holds b2 . b3 = cod b3 ) implies b1 = b2 )
func ID c
1 -> Function of a
1,
Morphs a
1 means :
Def22:
:: RINGCAT1:def 22
for b
1 being
Element of a
1 holds a
2 . b
1 = ID b
1;
existence
ex b1 being Function of c1, Morphs c1 st
for b2 being Element of c1 holds b1 . b2 = ID b2
uniqueness
for b1, b2 being Function of c1, Morphs c1 holds
( ( for b3 being Element of c1 holds b1 . b3 = ID b3 ) & ( for b3 being Element of c1 holds b2 . b3 = ID b3 ) implies b1 = b2 )
end;
:: deftheorem Def20 defines dom RINGCAT1:def 20 :
:: deftheorem Def21 defines cod RINGCAT1:def 21 :
:: deftheorem Def22 defines ID RINGCAT1:def 22 :
theorem Th24: :: RINGCAT1:24
theorem Th25: :: RINGCAT1:25
definition
let c
1 be
Ring_DOMAIN;
func comp c
1 -> PartFunc of
[:(Morphs a1),(Morphs a1):],
Morphs a
1 means :
Def23:
:: RINGCAT1:def 23
( ( for b
1, b
2 being
Element of
Morphs a
1 holds
(
[b1,b2] in dom a
2 iff
dom b
1 = cod b
2 ) ) & ( for b
1, b
2 being
Element of
Morphs a
1 holds
(
[b1,b2] in dom a
2 implies a
2 . [b1,b2] = b
1 * b
2 ) ) );
existence
ex b1 being PartFunc of [:(Morphs c1),(Morphs c1):], Morphs c1 st
( ( for b2, b3 being Element of Morphs c1 holds
( [b2,b3] in dom b1 iff dom b2 = cod b3 ) ) & ( for b2, b3 being Element of Morphs c1 holds
( [b2,b3] in dom b1 implies b1 . [b2,b3] = b2 * b3 ) ) )
uniqueness
for b1, b2 being PartFunc of [:(Morphs c1),(Morphs c1):], Morphs c1 holds
( ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b1 iff dom b3 = cod b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b1 implies b1 . [b3,b4] = b3 * b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b2 iff dom b3 = cod b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b2 implies b2 . [b3,b4] = b3 * b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def23 defines comp RINGCAT1:def 23 :
definition
let c
1 be
Universe;
func RingCat c
1 -> CatStr equals :: RINGCAT1:def 24
CatStr(#
(RingObjects a1),
(Morphs (RingObjects a1)),
(dom (RingObjects a1)),
(cod (RingObjects a1)),
(comp (RingObjects a1)),
(ID (RingObjects a1)) #);
coherence
CatStr(# (RingObjects c1),(Morphs (RingObjects c1)),(dom (RingObjects c1)),(cod (RingObjects c1)),(comp (RingObjects c1)),(ID (RingObjects c1)) #) is CatStr
;
end;
:: deftheorem Def24 defines RingCat RINGCAT1:def 24 :
theorem Th26: :: RINGCAT1:26
theorem Th27: :: RINGCAT1:27
theorem Th28: :: RINGCAT1:28
theorem Th29: :: RINGCAT1:29
theorem Th30: :: RINGCAT1:30
Lemma49:
for b1 being Universe
for b2, b3 being Morphism of (RingCat b1) holds
( dom b3 = cod b2 implies ( dom (b3 * b2) = dom b2 & cod (b3 * b2) = cod b3 ) )
Lemma50:
for b1 being Universe
for b2, b3, b4 being Morphism of (RingCat b1) holds
( dom b4 = cod b3 & dom b3 = cod b2 implies b4 * (b3 * b2) = (b4 * b3) * b2 )
Lemma51:
for b1 being Universe
for b2 being Object of (RingCat b1) holds
( dom (id b2) = b2 & cod (id b2) = b2 & ( for b3 being Morphism of (RingCat b1) holds
( cod b3 = b2 implies (id b2) * b3 = b3 ) ) & ( for b3 being Morphism of (RingCat b1) holds
( dom b3 = b2 implies b3 * (id b2) = b3 ) ) )