:: FINSOP_1 semantic presentation
:: deftheorem Def1 defines "**" FINSOP_1:def 1 :
theorem Th1: :: FINSOP_1:1
canceled;
theorem Th2: :: FINSOP_1:2
theorem Th3: :: FINSOP_1:3
Lemma83:
for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F >= 1 & g is associative & g is commutative holds
g "**" F = g $$ (dom F),((NAT --> (the_unity_wrt g)) +* F)
Lemma84:
for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 0 & g has_a_unity & g is associative & g is commutative holds
g "**" F = g $$ (dom F),((NAT --> (the_unity_wrt g)) +* F)
theorem Th4: :: FINSOP_1:4
Lemma85:
for D being non empty set
for g being BinOp of D st g has_a_unity holds
g "**" (<*> D) = the_unity_wrt g
Lemma86:
for D being non empty set
for d being Element of D
for g being BinOp of D holds g "**" <*d*> = d
Lemma87:
for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st len F >= 1 holds
g "**" (F ^ <*d*>) = g . (g "**" F),d
Lemma88:
for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st g has_a_unity & len F = 0 holds
g "**" (F ^ <*d*>) = g . (g "**" F),d
theorem Th5: :: FINSOP_1:5
theorem Th6: :: FINSOP_1:6
theorem Th7: :: FINSOP_1:7
Lemma92:
for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st g is associative & g is commutative holds
for f being Permutation of dom F st len F >= 1 & len F = len G & ( for i being Element of NAT st i in dom G holds
G . i = F . (f . i) ) holds
g "**" F = g "**" G
Lemma179:
for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D
for P being Permutation of dom F st g has_a_unity & len F = 0 & G = F * P holds
g "**" F = g "**" G
theorem Th8: :: FINSOP_1:8
Lemma181:
for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st g is associative & g is commutative & F is one-to-one & G is one-to-one & rng F = rng G & len F >= 1 holds
g "**" F = g "**" G
Lemma182:
for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st len F = 0 & g has_a_unity & F is one-to-one & G is one-to-one & rng F = rng G holds
g "**" F = g "**" G
theorem Th9: :: FINSOP_1:9
Lemma183:
for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 1 holds
g "**" F = F . 1
Lemma184:
for D being non empty set
for F, G, H being FinSequence of D
for g being BinOp of D st g is associative & g is commutative & len F >= 1 & len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = g . (F . k),(G . k) ) holds
g "**" H = g . (g "**" F),(g "**" G)
Lemma185:
for D being non empty set
for F, G, H being FinSequence of D
for g being BinOp of D st g has_a_unity & len F = 0 & len F = len G & len F = len H holds
g "**" F = g . (g "**" G),(g "**" H)
theorem Th10: :: FINSOP_1:10
theorem Th11: :: FINSOP_1:11
theorem Th12: :: FINSOP_1:12
theorem Th13: :: FINSOP_1:13
theorem Th14: :: FINSOP_1:14
theorem Th15: :: FINSOP_1:15
theorem Th16: :: FINSOP_1:16
theorem Th17: :: FINSOP_1:17
theorem Th18: :: FINSOP_1:18
theorem Th19: :: FINSOP_1:19
theorem Th20: :: FINSOP_1:20
theorem Th21: :: FINSOP_1:21
theorem Th22: :: FINSOP_1:22