:: VECTSP_2 semantic presentation
:: deftheorem Def1 VECTSP_2:def 1 :
canceled;
:: deftheorem Def2 defines well-unital VECTSP_2:def 2 :
Lemma19:
for L being non empty multLoopStr st L is well-unital holds
L is unital
Lemma23:
for L being non empty multLoopStr st L is well-unital holds
1_ L = 1. L
theorem Th1: :: VECTSP_2:1
:: deftheorem Def3 VECTSP_2:def 3 :
canceled;
:: deftheorem Def4 VECTSP_2:def 4 :
canceled;
:: deftheorem Def5 defines domRing-like VECTSP_2:def 5 :
theorem Th2: :: VECTSP_2:2
canceled;
theorem Th3: :: VECTSP_2:3
canceled;
theorem Th4: :: VECTSP_2:4
canceled;
theorem Th5: :: VECTSP_2:5
canceled;
theorem Th6: :: VECTSP_2:6
canceled;
theorem Th7: :: VECTSP_2:7
canceled;
theorem Th8: :: VECTSP_2:8
canceled;
theorem Th9: :: VECTSP_2:9
canceled;
theorem Th10: :: VECTSP_2:10
canceled;
theorem Th11: :: VECTSP_2:11
canceled;
theorem Th12: :: VECTSP_2:12
canceled;
theorem Th13: :: VECTSP_2:13
theorem Th14: :: VECTSP_2:14
canceled;
theorem Th15: :: VECTSP_2:15
canceled;
theorem Th16: :: VECTSP_2:16
Lemma30:
for R being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for x, y, z being Scalar of R st x + y = z holds
x = z - y
Lemma31:
for R being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for x, z, y being Scalar of R st x = z - y holds
x + y = z
theorem Th17: :: VECTSP_2:17
canceled;
theorem Th18: :: VECTSP_2:18
canceled;
theorem Th19: :: VECTSP_2:19
canceled;
theorem Th20: :: VECTSP_2:20
canceled;
theorem Th21: :: VECTSP_2:21
canceled;
theorem Th22: :: VECTSP_2:22
theorem Th23: :: VECTSP_2:23
canceled;
theorem Th24: :: VECTSP_2:24
canceled;
theorem Th25: :: VECTSP_2:25
canceled;
theorem Th26: :: VECTSP_2:26
canceled;
theorem Th27: :: VECTSP_2:27
canceled;
theorem Th28: :: VECTSP_2:28
canceled;
theorem Th29: :: VECTSP_2:29
canceled;
theorem Th30: :: VECTSP_2:30
canceled;
theorem Th31: :: VECTSP_2:31
canceled;
theorem Th32: :: VECTSP_2:32
canceled;
theorem Th33: :: VECTSP_2:33
canceled;
theorem Th34: :: VECTSP_2:34
theorem Th35: :: VECTSP_2:35
canceled;
theorem Th36: :: VECTSP_2:36
canceled;
theorem Th37: :: VECTSP_2:37
canceled;
theorem Th38: :: VECTSP_2:38
theorem Th39: :: VECTSP_2:39
theorem Th40: :: VECTSP_2:40
theorem Th41: :: VECTSP_2:41
theorem Th42: :: VECTSP_2:42
:: deftheorem Def6 VECTSP_2:def 6 :
canceled;
:: deftheorem Def7 defines " VECTSP_2:def 7 :
:: deftheorem Def8 defines / VECTSP_2:def 8 :
theorem Th43: :: VECTSP_2:43
theorem Th44: :: VECTSP_2:44
canceled;
theorem Th45: :: VECTSP_2:45
theorem Th46: :: VECTSP_2:46
theorem Th47: :: VECTSP_2:47
theorem Th48: :: VECTSP_2:48
theorem Th49: :: VECTSP_2:49
theorem Th50: :: VECTSP_2:50
theorem Th51: :: VECTSP_2:51
theorem Th52: :: VECTSP_2:52
theorem Th53: :: VECTSP_2:53
theorem Th54: :: VECTSP_2:54
theorem Th55: :: VECTSP_2:55
theorem Th56: :: VECTSP_2:56
theorem Th57: :: VECTSP_2:57
registration
let FS1 be
1-sorted ,
FS2 be
1-sorted ;
let A be non
empty set ;
let a be
BinOp of
A;
let Z be
Element of
A;
let l be
Function of
[:the carrier of FS1,A:],
A;
let r be
Function of
[:A,the carrier of FS2:],
A;
cluster BiModStr(#
a3,
a4,
a5,
a6,
a7 #)
-> non
empty ;
coherence
not BiModStr(# A,a,Z,l,r #) is empty
by STRUCT_0:def 1;
end;
:: deftheorem Def9 defines AbGr VECTSP_2:def 9 :
deffunc H1( Ring) -> VectSpStr of a1 = VectSpStr(# the carrier of a1,the add of a1,the Zero of a1,the mult of a1 #);
Lemma66:
for R being Ring holds
( H1(R) is Abelian & H1(R) is add-associative & H1(R) is right_zeroed & H1(R) is right_complementable )
:: deftheorem Def10 VECTSP_2:def 10 :
canceled;
:: deftheorem Def11 defines LeftModule VECTSP_2:def 11 :
deffunc H2( Ring) -> RightModStr of a1 = RightModStr(# the carrier of a1,the add of a1,the Zero of a1,the mult of a1 #);
Lemma69:
for R being Ring holds
( H2(R) is Abelian & H2(R) is add-associative & H2(R) is right_zeroed & H2(R) is right_complementable )
:: deftheorem Def12 VECTSP_2:def 12 :
canceled;
:: deftheorem Def13 VECTSP_2:def 13 :
canceled;
:: deftheorem Def14 defines RightModule VECTSP_2:def 14 :
:: deftheorem Def15 defines * VECTSP_2:def 15 :
:: deftheorem Def16 VECTSP_2:def 16 :
canceled;
:: deftheorem Def17 defines op1 VECTSP_2:def 17 :
:: deftheorem Def18 defines op0 VECTSP_2:def 18 :
deffunc H3( Ring, Ring) -> BiModStr of a1,a2 = BiModStr(# {{} },op2 ,op0 ,(pr2 the carrier of a1,{{} }),(pr1 {{} },the carrier of a2) #);
Lemma72:
for R1, R2 being Ring holds
( H3(R1,R2) is Abelian & H3(R1,R2) is add-associative & H3(R1,R2) is right_zeroed & H3(R1,R2) is right_complementable )
definition
let R1 be
Ring;
let R2 be
Ring;
canceled;canceled;func BiModule c1,
c2 -> non
empty Abelian add-associative right_zeroed right_complementable strict BiModStr of
a1,
a2 equals :: VECTSP_2:def 21
BiModStr(#
{{} },
op2 ,
op0 ,
(pr2 the carrier of R1,{{} }),
(pr1 {{} },the carrier of R2) #);
coherence
BiModStr(# {{} },op2 ,op0 ,(pr2 the carrier of R1,{{} }),(pr1 {{} },the carrier of R2) #) is non empty Abelian add-associative right_zeroed right_complementable strict BiModStr of R1,R2
by Lemma23;
end;
:: deftheorem Def19 VECTSP_2:def 19 :
canceled;
:: deftheorem Def20 VECTSP_2:def 20 :
canceled;
:: deftheorem Def21 defines BiModule VECTSP_2:def 21 :
theorem Th58: :: VECTSP_2:58
canceled;
theorem Th59: :: VECTSP_2:59
canceled;
theorem Th60: :: VECTSP_2:60
canceled;
theorem Th61: :: VECTSP_2:61
canceled;
theorem Th62: :: VECTSP_2:62
canceled;
theorem Th63: :: VECTSP_2:63
canceled;
theorem Th64: :: VECTSP_2:64
canceled;
theorem Th65: :: VECTSP_2:65
canceled;
theorem Th66: :: VECTSP_2:66
canceled;
theorem Th67: :: VECTSP_2:67
canceled;
theorem Th68: :: VECTSP_2:68
canceled;
theorem Th69: :: VECTSP_2:69
canceled;
theorem Th70: :: VECTSP_2:70
canceled;
theorem Th71: :: VECTSP_2:71
Lemma80:
for R being Ring holds LeftModule R is VectSp-like
theorem Th72: :: VECTSP_2:72
canceled;
theorem Th73: :: VECTSP_2:73
canceled;
theorem Th74: :: VECTSP_2:74
canceled;
theorem Th75: :: VECTSP_2:75
canceled;
theorem Th76: :: VECTSP_2:76
canceled;
theorem Th77: :: VECTSP_2:77
:: deftheorem Def22 VECTSP_2:def 22 :
canceled;
:: deftheorem Def23 defines RightMod-like VECTSP_2:def 23 :
Lemma84:
for R being Ring holds RightModule R is RightMod-like
Lemma85:
for R1, R2 being Ring
for x, y being Scalar of R1
for p, q being Scalar of R2
for v, w being Vector of (BiModule R1,R2) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1. R2) = v & x * (v * p) = (x * v) * p )
:: deftheorem Def24 defines BiMod-like VECTSP_2:def 24 :
theorem Th78: :: VECTSP_2:78
canceled;
theorem Th79: :: VECTSP_2:79
canceled;
theorem Th80: :: VECTSP_2:80
canceled;
theorem Th81: :: VECTSP_2:81
canceled;
theorem Th82: :: VECTSP_2:82
canceled;
theorem Th83: :: VECTSP_2:83
theorem Th84: :: VECTSP_2:84
theorem Th85: :: VECTSP_2:85