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Theorem ablfac1b 16576
Description: Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablfac1.b  |-  B  =  ( Base `  G
)
ablfac1.o  |-  O  =  ( od `  G
)
ablfac1.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac1.g  |-  ( ph  ->  G  e.  Abel )
ablfac1.f  |-  ( ph  ->  B  e.  Fin )
ablfac1.1  |-  ( ph  ->  A  C_  Prime )
Assertion
Ref Expression
ablfac1b  |-  ( ph  ->  G dom DProd  S )
Distinct variable groups:    x, p, B    ph, p, x    A, p, x    O, p, x    G, p, x
Allowed substitution hints:    S( x, p)

Proof of Theorem ablfac1b
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . 2  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2443 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2443 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 ablfac1.g . . 3  |-  ( ph  ->  G  e.  Abel )
5 ablgrp 16287 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
64, 5syl 16 . 2  |-  ( ph  ->  G  e.  Grp )
7 ablfac1.1 . . 3  |-  ( ph  ->  A  C_  Prime )
8 nnex 10333 . . . . 5  |-  NN  e.  _V
9 prmnn 13771 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
109ssriv 3365 . . . . 5  |-  Prime  C_  NN
118, 10ssexi 4442 . . . 4  |-  Prime  e.  _V
1211ssex 4441 . . 3  |-  ( A 
C_  Prime  ->  A  e.  _V )
137, 12syl 16 . 2  |-  ( ph  ->  A  e.  _V )
144adantr 465 . . . 4  |-  ( (
ph  /\  p  e.  A )  ->  G  e.  Abel )
157sselda 3361 . . . . . . 7  |-  ( (
ph  /\  p  e.  A )  ->  p  e.  Prime )
1615, 9syl 16 . . . . . 6  |-  ( (
ph  /\  p  e.  A )  ->  p  e.  NN )
17 ablfac1.b . . . . . . . . . . 11  |-  B  =  ( Base `  G
)
1817grpbn0 15572 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  B  =/=  (/) )
196, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  B  =/=  (/) )
20 ablfac1.f . . . . . . . . . 10  |-  ( ph  ->  B  e.  Fin )
21 hashnncl 12139 . . . . . . . . . 10  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2319, 22mpbird 232 . . . . . . . 8  |-  ( ph  ->  ( # `  B
)  e.  NN )
2423adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  A )  ->  ( # `
 B )  e.  NN )
2515, 24pccld 13922 . . . . . 6  |-  ( (
ph  /\  p  e.  A )  ->  (
p  pCnt  ( # `  B
) )  e.  NN0 )
2616, 25nnexpcld 12034 . . . . 5  |-  ( (
ph  /\  p  e.  A )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  NN )
2726nnzd 10751 . . . 4  |-  ( (
ph  /\  p  e.  A )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  ZZ )
28 ablfac1.o . . . . 5  |-  O  =  ( od `  G
)
2928, 17oddvdssubg 16342 . . . 4  |-  ( ( G  e.  Abel  /\  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  (SubGrp `  G
) )
3014, 27, 29syl2anc 661 . . 3  |-  ( (
ph  /\  p  e.  A )  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  (SubGrp `  G
) )
31 ablfac1.s . . 3  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
3230, 31fmptd 5872 . 2  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
334adantr 465 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  ->  G  e.  Abel )
3432adantr 465 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  ->  S : A --> (SubGrp `  G ) )
35 simpr1 994 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
a  e.  A )
3634, 35ffvelrnd 5849 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  a
)  e.  (SubGrp `  G ) )
37 simpr2 995 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
b  e.  A )
3834, 37ffvelrnd 5849 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  b
)  e.  (SubGrp `  G ) )
391, 33, 36, 38ablcntzd 16344 . 2  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  a
)  C_  ( (Cntz `  G ) `  ( S `  b )
) )
40 id 22 . . . . . . . . . 10  |-  ( p  =  a  ->  p  =  a )
41 oveq1 6103 . . . . . . . . . 10  |-  ( p  =  a  ->  (
p  pCnt  ( # `  B
) )  =  ( a  pCnt  ( # `  B
) ) )
4240, 41oveq12d 6114 . . . . . . . . 9  |-  ( p  =  a  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( a ^ (
a  pCnt  ( # `  B
) ) ) )
4342breq2d 4309 . . . . . . . 8  |-  ( p  =  a  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) ) )
4443rabbidv 2969 . . . . . . 7  |-  ( p  =  a  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) } )
45 fvex 5706 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
4617, 45eqeltri 2513 . . . . . . . 8  |-  B  e. 
_V
4746rabex 4448 . . . . . . 7  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
4844, 31, 47fvmpt3i 5783 . . . . . 6  |-  ( a  e.  A  ->  ( S `  a )  =  { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
4948adantl 466 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( S `  a )  =  { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
50 eqimss 3413 . . . . 5  |-  ( ( S `  a )  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  ->  ( S `  a )  C_  { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) } )
5149, 50syl 16 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( S `  a )  C_ 
{ x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
524adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  G  e.  Abel )
53 eqid 2443 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
5453subgacs 15721 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
55 acsmre 14595 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5652, 5, 54, 554syl 21 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
57 df-ima 4858 . . . . . . 7  |-  ( S
" ( A  \  { a } ) )  =  ran  ( S  |`  ( A  \  { a } ) )
587sselda 3361 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  Prime )
5958ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  e.  Prime )
6023ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  e.  NN )
61 pcdvds 13935 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
6259, 60, 61syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
637ad3antrrr 729 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  A  C_ 
Prime )
64 eldifi 3483 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( A  \  { a } )  ->  p  e.  A
)
6564ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  A )
6663, 65sseldd 3362 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  Prime )
67 pcdvds 13935 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
6866, 60, 67syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
69 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a ^ ( a  pCnt  (
# `  B )
) )  =  ( a ^ ( a 
pCnt  ( # `  B
) ) )
70 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) )
7117, 28, 31, 4, 20, 7, 69, 70ablfac1lem 16574 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  A )  ->  (
( ( a ^
( a  pCnt  ( # `
 B ) ) )  e.  NN  /\  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) )  e.  NN )  /\  (
( a ^ (
a  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )  =  1  /\  ( # `
 B )  =  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) ) )
7271simp1d 1000 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  A )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
7372simpld 459 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  A )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  NN )
7473ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  NN )
7574nnzd 10751 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  ZZ )
7666, 9syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  NN )
7766, 60pccld 13922 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p  pCnt  ( # `  B
) )  e.  NN0 )
7876, 77nnexpcld 12034 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  NN )
7978nnzd 10751 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  ZZ )
8060nnzd 10751 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  e.  ZZ )
81 eldifsni 4006 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( A  \  { a } )  ->  p  =/=  a
)
8281ad2antlr 726 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  =/=  a )
8382necomd 2700 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  =/=  p )
84 prmrp 13792 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  Prime  /\  p  e.  Prime )  ->  (
( a  gcd  p
)  =  1  <->  a  =/=  p ) )
8559, 66, 84syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a  gcd  p
)  =  1  <->  a  =/=  p ) )
8683, 85mpbird 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a  gcd  p )  =  1 )
87 prmz 13772 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  Prime  ->  a  e.  ZZ )
8859, 87syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  e.  ZZ )
89 prmz 13772 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  Prime  ->  p  e.  ZZ )
9066, 89syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  ZZ )
9159, 60pccld 13922 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a  pCnt  ( # `  B
) )  e.  NN0 )
92 rpexp12i 13813 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  e.  ZZ  /\  p  e.  ZZ  /\  (
( a  pCnt  ( # `
 B ) )  e.  NN0  /\  (
p  pCnt  ( # `  B
) )  e.  NN0 ) )  ->  (
( a  gcd  p
)  =  1  -> 
( ( a ^
( a  pCnt  ( # `
 B ) ) )  gcd  ( p ^ ( p  pCnt  (
# `  B )
) ) )  =  1 ) )
9388, 90, 91, 77, 92syl112anc 1222 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a  gcd  p
)  =  1  -> 
( ( a ^
( a  pCnt  ( # `
 B ) ) )  gcd  ( p ^ ( p  pCnt  (
# `  B )
) ) )  =  1 ) )
9486, 93mpd 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  gcd  ( p ^ (
p  pCnt  ( # `  B
) ) ) )  =  1 )
95 coprmdvds2 13794 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a ^
( a  pCnt  ( # `
 B ) ) )  e.  ZZ  /\  ( p ^ (
p  pCnt  ( # `  B
) ) )  e.  ZZ  /\  ( # `  B )  e.  ZZ )  /\  ( ( a ^ ( a  pCnt  (
# `  B )
) )  gcd  (
p ^ ( p 
pCnt  ( # `  B
) ) ) )  =  1 )  -> 
( ( ( a ^ ( a  pCnt  (
# `  B )
) )  ||  ( # `
 B )  /\  ( p ^ (
p  pCnt  ( # `  B
) ) )  ||  ( # `  B ) )  ->  ( (
a ^ ( a 
pCnt  ( # `  B
) ) )  x.  ( p ^ (
p  pCnt  ( # `  B
) ) ) ) 
||  ( # `  B
) ) )
9675, 79, 80, 94, 95syl31anc 1221 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( ( a ^
( a  pCnt  ( # `
 B ) ) )  ||  ( # `  B )  /\  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )  ->  ( (
a ^ ( a 
pCnt  ( # `  B
) ) )  x.  ( p ^ (
p  pCnt  ( # `  B
) ) ) ) 
||  ( # `  B
) ) )
9762, 68, 96mp2and 679 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  x.  ( p ^ (
p  pCnt  ( # `  B
) ) ) ) 
||  ( # `  B
) )
9871simp3d 1002 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  A )  ->  ( # `
 B )  =  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
9998ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  =  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
10097, 99breqtrd 4321 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  x.  ( p ^ (
p  pCnt  ( # `  B
) ) ) ) 
||  ( ( a ^ ( a  pCnt  (
# `  B )
) )  x.  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
10172simprd 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  A )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN )
102101ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN )
103102nnzd 10751 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )
10474nnne0d 10371 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  =/=  0 )
105 dvdscmulr 13566 . . . . . . . . . . . . . 14  |-  ( ( ( p ^ (
p  pCnt  ( # `  B
) ) )  e.  ZZ  /\  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ  /\  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  e.  ZZ  /\  ( a ^ (
a  pCnt  ( # `  B
) ) )  =/=  0 ) )  -> 
( ( ( a ^ ( a  pCnt  (
# `  B )
) )  x.  (
p ^ ( p 
pCnt  ( # `  B
) ) ) ) 
||  ( ( a ^ ( a  pCnt  (
# `  B )
) )  x.  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) )  <->  ( p ^ ( p  pCnt  (
# `  B )
) )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
10679, 103, 75, 104, 105syl112anc 1222 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( p ^ ( p  pCnt  (
# `  B )
) ) )  ||  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) )  <->  ( p ^ ( p  pCnt  (
# `  B )
) )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
107100, 106mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )
10817, 28odcl 16044 . . . . . . . . . . . . . . 15  |-  ( x  e.  B  ->  ( O `  x )  e.  NN0 )
109108adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( O `  x )  e.  NN0 )
110109nn0zd 10750 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( O `  x )  e.  ZZ )
111 dvdstr 13571 . . . . . . . . . . . . 13  |-  ( ( ( O `  x
)  e.  ZZ  /\  ( p ^ (
p  pCnt  ( # `  B
) ) )  e.  ZZ  /\  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )  ->  ( ( ( O `  x ) 
||  ( p ^
( p  pCnt  ( # `
 B ) ) )  /\  ( p ^ ( p  pCnt  (
# `  B )
) )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) )  -> 
( O `  x
)  ||  ( ( # `
 B )  / 
( a ^ (
a  pCnt  ( # `  B
) ) ) ) ) )
112110, 79, 103, 111syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) )  /\  ( p ^ (
p  pCnt  ( # `  B
) ) )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )  ->  ( O `  x )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
113107, 112mpan2d 674 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  ->  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) ) )
114113ss2rabdv 3438 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  {
a } ) )  ->  { x  e.  B  |  ( O `
 x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } 
C_  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
11547elpw 3871 . . . . . . . . . 10  |-  ( { x  e.  B  | 
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) ) }  e.  ~P { x  e.  B  |  ( O `  x )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) }  <->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } 
C_  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
116114, 115sylibr 212 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  {
a } ) )  ->  { x  e.  B  |  ( O `
 x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
11731reseq1i 5111 . . . . . . . . . 10  |-  ( S  |`  ( A  \  {
a } ) )  =  ( ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  |`  ( A  \  { a } ) )
118 difss 3488 . . . . . . . . . . 11  |-  ( A 
\  { a } )  C_  A
119 resmpt 5161 . . . . . . . . . . 11  |-  ( ( A  \  { a } )  C_  A  ->  ( ( p  e.  A  |->  { x  e.  B  |  ( O `
 x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  |`  ( A  \  { a } ) )  =  ( p  e.  ( A  \  { a } ) 
|->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } ) )
120118, 119ax-mp 5 . . . . . . . . . 10  |-  ( ( p  e.  A  |->  { x  e.  B  | 
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) ) } )  |`  ( A  \  {
a } ) )  =  ( p  e.  ( A  \  {
a } )  |->  { x  e.  B  | 
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) ) } )
121117, 120eqtri 2463 . . . . . . . . 9  |-  ( S  |`  ( A  \  {
a } ) )  =  ( p  e.  ( A  \  {
a } )  |->  { x  e.  B  | 
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) ) } )
122116, 121fmptd 5872 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( S  |`  ( A  \  { a } ) ) : ( A 
\  { a } ) --> ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
123 frn 5570 . . . . . . . 8  |-  ( ( S  |`  ( A  \  { a } ) ) : ( A 
\  { a } ) --> ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  ->  ran  ( S  |`  ( A  \  {
a } ) ) 
C_  ~P { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
124122, 123syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ran  ( S  |`  ( A 
\  { a } ) )  C_  ~P { x  e.  B  |  ( O `  x )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) } )
12557, 124syl5eqss 3405 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( S " ( A  \  { a } ) )  C_  ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
126 sspwuni 4261 . . . . . 6  |-  ( ( S " ( A 
\  { a } ) )  C_  ~P { x  e.  B  |  ( O `  x )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) }  <->  U. ( S " ( A  \  { a } ) )  C_  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
127125, 126sylib 196 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  U. ( S " ( A  \  { a } ) )  C_  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
128101nnzd 10751 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )
12928, 17oddvdssubg 16342 . . . . . 6  |-  ( ( G  e.  Abel  /\  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )  ->  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  e.  (SubGrp `  G
) )
13052, 128, 129syl2anc 661 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  e.  (SubGrp `  G
) )
1313mrcsscl 14563 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( A  \  { a } ) )  C_  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  /\  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( A 
\  { a } ) ) )  C_  { x  e.  B  | 
( O `  x
)  ||  ( ( # `
 B )  / 
( a ^ (
a  pCnt  ( # `  B
) ) ) ) } )
13256, 127, 130, 131syl3anc 1218 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( A 
\  { a } ) ) )  C_  { x  e.  B  | 
( O `  x
)  ||  ( ( # `
 B )  / 
( a ^ (
a  pCnt  ( # `  B
) ) ) ) } )
133 ss2in 3582 . . . 4  |-  ( ( ( S `  a
)  C_  { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  /\  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( A  \  { a } ) ) )  C_  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )  ->  ( ( S `  a )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( A  \  { a } ) ) ) ) 
C_  ( { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  i^i  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } ) )
13451, 132, 133syl2anc 661 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  (
( S `  a
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( A  \  { a } ) ) ) )  C_  ( { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) }  i^i  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } ) )
135 eqid 2443 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }
136 eqid 2443 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }
13771simp2d 1001 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )  =  1 )
138 eqid 2443 . . . . 5  |-  ( LSSum `  G )  =  (
LSSum `  G )
13917, 28, 135, 136, 52, 73, 101, 137, 98, 2, 138ablfacrp 16572 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  i^i  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )  =  { ( 0g `  G ) }  /\  ( { x  e.  B  | 
( O `  x
)  ||  ( a ^ ( a  pCnt  (
# `  B )
) ) }  ( LSSum `  G ) { x  e.  B  | 
( O `  x
)  ||  ( ( # `
 B )  / 
( a ^ (
a  pCnt  ( # `  B
) ) ) ) } )  =  B ) )
140139simpld 459 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) }  i^i  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )  =  { ( 0g `  G ) } )
141134, 140sseqtrd 3397 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( S `  a
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( A  \  { a } ) ) ) )  C_  { ( 0g `  G
) } )
1421, 2, 3, 6, 13, 32, 39, 141dmdprdd 16486 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   {crab 2724   _Vcvv 2977    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   {csn 3882   U.cuni 4096   class class class wbr 4297    e. cmpt 4355   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   Fincfn 7315   0cc0 9287   1c1 9288    x. cmul 9292    / cdiv 9998   NNcn 10327   NN0cn0 10584   ZZcz 10651   ^cexp 11870   #chash 12108    || cdivides 13540    gcd cgcd 13695   Primecprime 13768    pCnt cpc 13908   Basecbs 14179   0gc0g 14383  Moorecmre 14525  mrClscmrc 14526  ACScacs 14528   Grpcgrp 15415  SubGrpcsubg 15680  Cntzccntz 15838   odcod 16033   LSSumclsm 16138   Abelcabel 16283   DProd cdprd 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-eqg 15685  df-cntz 15840  df-od 16037  df-lsm 16140  df-cmn 16284  df-abl 16285  df-dprd 16482
This theorem is referenced by:  ablfac1c  16577  ablfac1eu  16579  ablfaclem2  16592  ablfaclem3  16593
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