:: MOD_4 semantic presentation
theorem Th1: :: MOD_4:1
:: deftheorem Def1 defines opp MOD_4:def 1 :
Lemma2:
for b1 being non empty doubleLoopStr
for b2, b3 being Scalar of b1
for b4, b5 being Scalar of (opp b1) holds
( b2 = b4 & b3 = b5 implies ( b2 + b3 = b4 + b5 & b2 * b3 = b5 * b4 ) )
by Th1;
Lemma3:
for b1 being non empty unital doubleLoopStr
for b2, b3 being Element of (opp b1) holds
( b3 = 1. b1 implies ( b2 * b3 = b2 & b3 * b2 = b2 ) )
Lemma4:
for b1 being non empty unital doubleLoopStr holds 1. b1 = 1. (opp b1)
Lemma6:
for b1 being non empty add-associative right_zeroed right_complementable doubleLoopStr
for b2, b3 being Scalar of b1
for b4, b5 being Scalar of (opp b1) holds
( b2 = b4 & b3 = b5 implies ( b2 + b3 = b4 + b5 & b2 * b3 = b5 * b4 & - b2 = - b4 ) )
theorem Th2: :: MOD_4:2
theorem Th3: :: MOD_4:3
Lemma8:
for b1 being non empty add-associative right_zeroed right_complementable doubleLoopStr
for b2, b3, b4, b5 being Scalar of b1
for b6, b7, b8, b9 being Scalar of (opp b1) holds
( b2 = b6 & b3 = b7 & b4 = b8 & b5 = b9 implies ( (b2 + b3) + b4 = (b6 + b7) + b8 & b2 + (b3 + b4) = b6 + (b7 + b8) & (b2 * b3) * b4 = b8 * (b7 * b6) & b2 * (b3 * b4) = (b8 * b7) * b6 & b2 * (b3 + b4) = (b7 + b8) * b6 & (b3 + b4) * b2 = b6 * (b7 + b8) & (b2 * b3) + (b4 * b5) = (b7 * b6) + (b9 * b8) ) )
theorem Th4: :: MOD_4:4
theorem Th5: :: MOD_4:5
theorem Th6: :: MOD_4:6
theorem Th7: :: MOD_4:7
Lemma12:
for b1 being non empty commutative doubleLoopStr
for b2, b3 being Element of b1 holds b2 * b3 = b3 * b2
;
theorem Th8: :: MOD_4:8
definition
let c
1 be non
empty doubleLoopStr ;
let c
2 be non
empty VectSpStr of c
1;
func opp c
2 -> strict RightModStr of
opp a
1 means :
Def2:
:: MOD_4:def 2
for b
1 being
Function of
[:the carrier of a2,the carrier of (opp a1):],the
carrier of a
2 holds
( b
1 = ~ the
lmult of a
2 implies a
3 = RightModStr(# the
carrier of a
2,the
add of a
2,the
Zero of a
2,b
1 #) );
existence
ex b1 being strict RightModStr of opp c1 st
for b2 being Function of [:the carrier of c2,the carrier of (opp c1):],the carrier of c2 holds
( b2 = ~ the lmult of c2 implies b1 = RightModStr(# the carrier of c2,the add of c2,the Zero of c2,b2 #) )
uniqueness
for b1, b2 being strict RightModStr of opp c1 holds
( ( for b3 being Function of [:the carrier of c2,the carrier of (opp c1):],the carrier of c2 holds
( b3 = ~ the lmult of c2 implies b1 = RightModStr(# the carrier of c2,the add of c2,the Zero of c2,b3 #) ) ) & ( for b3 being Function of [:the carrier of c2,the carrier of (opp c1):],the carrier of c2 holds
( b3 = ~ the lmult of c2 implies b2 = RightModStr(# the carrier of c2,the add of c2,the Zero of c2,b3 #) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines opp MOD_4:def 2 :
theorem Th9: :: MOD_4:9
:: deftheorem Def3 defines opp MOD_4:def 3 :
theorem Th10: :: MOD_4:10
definition
let c
1 be non
empty doubleLoopStr ;
let c
2 be non
empty RightModStr of c
1;
func opp c
2 -> strict VectSpStr of
opp a
1 means :
Def4:
:: MOD_4:def 4
for b
1 being
Function of
[:the carrier of (opp a1),the carrier of a2:],the
carrier of a
2 holds
( b
1 = ~ the
rmult of a
2 implies a
3 = VectSpStr(# the
carrier of a
2,the
add of a
2,the
Zero of a
2,b
1 #) );
existence
ex b1 being strict VectSpStr of opp c1 st
for b2 being Function of [:the carrier of (opp c1),the carrier of c2:],the carrier of c2 holds
( b2 = ~ the rmult of c2 implies b1 = VectSpStr(# the carrier of c2,the add of c2,the Zero of c2,b2 #) )
uniqueness
for b1, b2 being strict VectSpStr of opp c1 holds
( ( for b3 being Function of [:the carrier of (opp c1),the carrier of c2:],the carrier of c2 holds
( b3 = ~ the rmult of c2 implies b1 = VectSpStr(# the carrier of c2,the add of c2,the Zero of c2,b3 #) ) ) & ( for b3 being Function of [:the carrier of (opp c1),the carrier of c2:],the carrier of c2 holds
( b3 = ~ the rmult of c2 implies b2 = VectSpStr(# the carrier of c2,the add of c2,the Zero of c2,b3 #) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines opp MOD_4:def 4 :
theorem Th11: :: MOD_4:11
canceled;
theorem Th12: :: MOD_4:12
:: deftheorem Def5 defines opp MOD_4:def 5 :
theorem Th13: :: MOD_4:13
theorem Th14: :: MOD_4:14
canceled;
theorem Th15: :: MOD_4:15
theorem Th16: :: MOD_4:16
theorem Th17: :: MOD_4:17
theorem Th18: :: MOD_4:18
theorem Th19: :: MOD_4:19
theorem Th20: :: MOD_4:20
theorem Th21: :: MOD_4:21
theorem Th22: :: MOD_4:22
theorem Th23: :: MOD_4:23
theorem Th24: :: MOD_4:24
theorem Th25: :: MOD_4:25
theorem Th26: :: MOD_4:26
:: deftheorem Def6 defines antilinear MOD_4:def 6 :
:: deftheorem Def7 defines monomorphism MOD_4:def 7 :
:: deftheorem Def8 defines antimonomorphism MOD_4:def 8 :
:: deftheorem Def9 defines epimorphism MOD_4:def 9 :
:: deftheorem Def10 defines antiepimorphism MOD_4:def 10 :
:: deftheorem Def11 defines isomorphism MOD_4:def 11 :
:: deftheorem Def12 defines antiisomorphism MOD_4:def 12 :
:: deftheorem Def13 defines endomorphism MOD_4:def 13 :
:: deftheorem Def14 defines antiendomorphism MOD_4:def 14 :
:: deftheorem Def15 defines automorphism MOD_4:def 15 :
:: deftheorem Def16 defines antiautomorphism MOD_4:def 16 :
theorem Th27: :: MOD_4:27
theorem Th28: :: MOD_4:28
Lemma41:
for b1 being non empty doubleLoopStr holds
( ( for b2, b3 being Scalar of b1 holds (id b1) . (b2 + b3) = ((id b1) . b2) + ((id b1) . b3) ) & ( for b2, b3 being Scalar of b1 holds (id b1) . (b2 * b3) = ((id b1) . b2) * ((id b1) . b3) ) & (id b1) . (1. b1) = 1. b1 & id b1 is one-to-one & rng (id b1) = the carrier of b1 )
theorem Th29: :: MOD_4:29
Lemma42:
for b1, b2 being Ring
for b3 being Function of b1,b2 holds
( b3 is linear implies b3 . (0. b1) = 0. b2 )
Lemma43:
for b1, b2 being Ring
for b3 being Function of b2,b1
for b4 being Scalar of b2 holds
( b3 is linear implies b3 . (- b4) = - (b3 . b4) )
Lemma44:
for b1, b2 being Ring
for b3 being Function of b2,b1
for b4, b5 being Scalar of b2 holds
( b3 is linear implies b3 . (b4 - b5) = (b3 . b4) - (b3 . b5) )
theorem Th30: :: MOD_4:30
Lemma45:
for b1, b2 being Ring
for b3 being Function of b1,b2 holds
( b3 is antilinear implies b3 . (0. b1) = 0. b2 )
Lemma46:
for b1, b2 being Ring
for b3 being Function of b2,b1
for b4 being Scalar of b2 holds
( b3 is antilinear implies b3 . (- b4) = - (b3 . b4) )
Lemma47:
for b1, b2 being Ring
for b3 being Function of b2,b1
for b4, b5 being Scalar of b2 holds
( b3 is antilinear implies b3 . (b4 - b5) = (b3 . b4) - (b3 . b5) )
theorem Th31: :: MOD_4:31
theorem Th32: :: MOD_4:32
theorem Th33: :: MOD_4:33
:: deftheorem Def17 defines opp MOD_4:def 17 :
theorem Th34: :: MOD_4:34
theorem Th35: :: MOD_4:35
theorem Th36: :: MOD_4:36
theorem Th37: :: MOD_4:37
theorem Th38: :: MOD_4:38
theorem Th39: :: MOD_4:39
theorem Th40: :: MOD_4:40
theorem Th41: :: MOD_4:41
theorem Th42: :: MOD_4:42
theorem Th43: :: MOD_4:43
theorem Th44: :: MOD_4:44
theorem Th45: :: MOD_4:45
theorem Th46: :: MOD_4:46
Lemma54:
for b1, b2 being AddGroup
for b3, b4 being Element of b1 holds (ZeroMap b1,b2) . (b3 + b4) = ((ZeroMap b1,b2) . b3) + ((ZeroMap b1,b2) . b4)
:: deftheorem Def18 defines Homomorphism MOD_4:def 18 :
:: deftheorem Def19 defines monomorphism MOD_4:def 19 :
:: deftheorem Def20 defines epimorphism MOD_4:def 20 :
:: deftheorem Def21 defines isomorphism MOD_4:def 21 :
Lemma57:
for b1 being AddGroup
for b2 being Element of b1 holds (id b1) . b2 = b2
by FUNCT_1:35;
Lemma58:
for b1 being AddGroup holds
( ( for b2, b3 being Element of b1 holds (id b1) . (b2 + b3) = ((id b1) . b2) + ((id b1) . b3) ) & id b1 is one-to-one & rng (id b1) = the carrier of b1 )
Lemma59:
for b1, b2 being AddGroup
for b3 being Homomorphism of b1,b2 holds b3 . (0. b1) = 0. b2
Lemma60:
for b1, b2 being AddGroup
for b3 being Homomorphism of b2,b1
for b4 being Element of b2 holds b3 . (- b4) = - (b3 . b4)
Lemma61:
for b1, b2 being AddGroup
for b3 being Homomorphism of b2,b1
for b4, b5 being Element of b2 holds b3 . (b4 - b5) = (b3 . b4) - (b3 . b5)
theorem Th47: :: MOD_4:47
canceled;
theorem Th48: :: MOD_4:48
theorem Th49: :: MOD_4:49
Lemma62:
for b1, b2 being Ring
for b3 being LeftMod of b1
for b4 being LeftMod of b2
for b5, b6 being Vector of b3 holds (ZeroMap b3,b4) . (b5 + b6) = ((ZeroMap b3,b4) . b5) + ((ZeroMap b3,b4) . b6)
Lemma63:
for b1, b2 being Ring
for b3 being Function of b2,b1
for b4 being LeftMod of b2
for b5 being LeftMod of b1
for b6 being Scalar of b2
for b7 being Vector of b4 holds (ZeroMap b4,b5) . (b6 * b7) = (b3 . b6) * ((ZeroMap b4,b5) . b7)
:: deftheorem Def22 MOD_4:def 22 :
canceled;
:: deftheorem Def23 defines Homomorphism MOD_4:def 23 :
theorem Th50: :: MOD_4:50
:: deftheorem Def24 defines is_monomorphism_wrp MOD_4:def 24 :
:: deftheorem Def25 defines is_epimorphism_wrp MOD_4:def 25 :
:: deftheorem Def26 defines is_isomorphism_wrp MOD_4:def 26 :
:: deftheorem Def27 defines is_automorphism_wrp MOD_4:def 27 :
theorem Th51: :: MOD_4:51
:: deftheorem Def28 defines monomorphism MOD_4:def 28 :
:: deftheorem Def29 defines epimorphism MOD_4:def 29 :
:: deftheorem Def30 defines isomorphism MOD_4:def 30 :
:: deftheorem Def31 defines automorphism MOD_4:def 31 :