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Theorem ablfac1b 15321
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 2296 . 2  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2296 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2296 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 ablfac1.g . . 3  |-  ( ph  ->  G  e.  Abel )
5 ablgrp 15110 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
64, 5syl 15 . 2  |-  ( ph  ->  G  e.  Grp )
7 ablfac1.1 . . 3  |-  ( ph  ->  A  C_  Prime )
8 nnex 9768 . . . . 5  |-  NN  e.  _V
9 prmnn 12777 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
109ssriv 3197 . . . . 5  |-  Prime  C_  NN
118, 10ssexi 4175 . . . 4  |-  Prime  e.  _V
1211ssex 4174 . . 3  |-  ( A 
C_  Prime  ->  A  e.  _V )
137, 12syl 15 . 2  |-  ( ph  ->  A  e.  _V )
144adantr 451 . . . 4  |-  ( (
ph  /\  p  e.  A )  ->  G  e.  Abel )
157sselda 3193 . . . . . . 7  |-  ( (
ph  /\  p  e.  A )  ->  p  e.  Prime )
1615, 9syl 15 . . . . . 6  |-  ( (
ph  /\  p  e.  A )  ->  p  e.  NN )
17 ablfac1.b . . . . . . . . . . 11  |-  B  =  ( Base `  G
)
1817grpbn0 14527 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  B  =/=  (/) )
196, 18syl 15 . . . . . . . . 9  |-  ( ph  ->  B  =/=  (/) )
20 ablfac1.f . . . . . . . . . 10  |-  ( ph  ->  B  e.  Fin )
21 hashnncl 11370 . . . . . . . . . 10  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
2220, 21syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
2319, 22mpbird 223 . . . . . . . 8  |-  ( ph  ->  ( # `  B
)  e.  NN )
2423adantr 451 . . . . . . 7  |-  ( (
ph  /\  p  e.  A )  ->  ( # `
 B )  e.  NN )
2515, 24pccld 12919 . . . . . 6  |-  ( (
ph  /\  p  e.  A )  ->  (
p  pCnt  ( # `  B
) )  e.  NN0 )
2616, 25nnexpcld 11282 . . . . 5  |-  ( (
ph  /\  p  e.  A )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  NN )
2726nnzd 10132 . . . 4  |-  ( (
ph  /\  p  e.  A )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  ZZ )
28 ablfac1.o . . . . 5  |-  O  =  ( od `  G
)
2928, 17oddvdssubg 15163 . . . 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 642 . . 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 5700 . 2  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
334adantr 451 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  ->  G  e.  Abel )
3432adantr 451 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  ->  S : A --> (SubGrp `  G ) )
35 simpr1 961 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
a  e.  A )
36 ffvelrn 5679 . . . 4  |-  ( ( S : A --> (SubGrp `  G )  /\  a  e.  A )  ->  ( S `  a )  e.  (SubGrp `  G )
)
3734, 35, 36syl2anc 642 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  a
)  e.  (SubGrp `  G ) )
38 simpr2 962 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
b  e.  A )
39 ffvelrn 5679 . . . 4  |-  ( ( S : A --> (SubGrp `  G )  /\  b  e.  A )  ->  ( S `  b )  e.  (SubGrp `  G )
)
4034, 38, 39syl2anc 642 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  b
)  e.  (SubGrp `  G ) )
411, 33, 37, 40ablcntzd 15165 . 2  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A  /\  a  =/=  b ) )  -> 
( S `  a
)  C_  ( (Cntz `  G ) `  ( S `  b )
) )
42 id 19 . . . . . . . . . 10  |-  ( p  =  a  ->  p  =  a )
43 oveq1 5881 . . . . . . . . . 10  |-  ( p  =  a  ->  (
p  pCnt  ( # `  B
) )  =  ( a  pCnt  ( # `  B
) ) )
4442, 43oveq12d 5892 . . . . . . . . 9  |-  ( p  =  a  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( a ^ (
a  pCnt  ( # `  B
) ) ) )
4544breq2d 4051 . . . . . . . 8  |-  ( p  =  a  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) ) )
4645rabbidv 2793 . . . . . . 7  |-  ( p  =  a  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) } )
47 fvex 5555 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
4817, 47eqeltri 2366 . . . . . . . 8  |-  B  e. 
_V
4948rabex 4181 . . . . . . 7  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
5046, 31, 49fvmpt3i 5621 . . . . . 6  |-  ( a  e.  A  ->  ( S `  a )  =  { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
5150adantl 452 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( S `  a )  =  { x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
52 eqimss 3243 . . . . 5  |-  ( ( S `  a )  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  ->  ( S `  a )  C_  { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) } )
5351, 52syl 15 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( S `  a )  C_ 
{ x  e.  B  |  ( O `  x )  ||  (
a ^ ( a 
pCnt  ( # `  B
) ) ) } )
544adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  G  e.  Abel )
5554, 5syl 15 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  G  e.  Grp )
56 eqid 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
5756subgacs 14668 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
58 acsmre 13570 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
5955, 57, 583syl 18 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
60 df-ima 4718 . . . . . . 7  |-  ( S
" ( A  \  { a } ) )  =  ran  ( S  |`  ( A  \  { a } ) )
617sselda 3193 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  Prime )
6261ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  e.  Prime )
6323ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  e.  NN )
64 pcdvds 12932 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
6562, 63, 64syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
667ad3antrrr 710 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  A  C_ 
Prime )
67 eldifi 3311 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  ( A  \  { a } )  ->  p  e.  A
)
6867ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  A )
6966, 68sseldd 3194 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  Prime )
70 pcdvds 12932 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
7169, 63, 70syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( # `  B ) )
72 eqid 2296 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a ^ ( a  pCnt  (
# `  B )
) )  =  ( a ^ ( a 
pCnt  ( # `  B
) ) )
73 eqid 2296 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) )
7417, 28, 31, 4, 20, 7, 72, 73ablfac1lem 15319 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
7574simp1d 967 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  A )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
7675simpld 445 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  A )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  NN )
7776ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  NN )
7877nnzd 10132 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  e.  ZZ )
7969, 9syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  NN )
8069, 63pccld 12919 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p  pCnt  ( # `  B
) )  e.  NN0 )
8179, 80nnexpcld 11282 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  NN )
8281nnzd 10132 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  e.  ZZ )
8363nnzd 10132 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  e.  ZZ )
84 eldifsni 3763 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( A  \  { a } )  ->  p  =/=  a
)
8584ad2antlr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  =/=  a )
8685necomd 2542 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  =/=  p )
87 prmrp 12796 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  Prime  /\  p  e.  Prime )  ->  (
( a  gcd  p
)  =  1  <->  a  =/=  p ) )
8862, 69, 87syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a  gcd  p
)  =  1  <->  a  =/=  p ) )
8986, 88mpbird 223 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a  gcd  p )  =  1 )
90 prmz 12778 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  Prime  ->  a  e.  ZZ )
9162, 90syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  a  e.  ZZ )
92 prmz 12778 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  Prime  ->  p  e.  ZZ )
9369, 92syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  p  e.  ZZ )
9462, 63pccld 12919 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a  pCnt  ( # `  B
) )  e.  NN0 )
95 rpexp12i 12817 . . . . . . . . . . . . . . . . . 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 ) )
9691, 93, 94, 80, 95syl112anc 1186 . . . . . . . . . . . . . . . . 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 ) )
9789, 96mpd 14 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  gcd  ( p ^ (
p  pCnt  ( # `  B
) ) ) )  =  1 )
98 coprmdvds2 12798 . . . . . . . . . . . . . . . 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
) ) )
9978, 82, 83, 97, 98syl31anc 1185 . . . . . . . . . . . . . . 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
) ) )
10065, 71, 99mp2and 660 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  x.  ( p ^ (
p  pCnt  ( # `  B
) ) ) ) 
||  ( # `  B
) )
10174simp3d 969 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  A )  ->  ( # `
 B )  =  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
102101ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( # `
 B )  =  ( ( a ^
( a  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) ) )
103100, 102breqtrd 4063 . . . . . . . . . . . . 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 ) ) ) ) ) )
10475simprd 449 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  A )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN )
105104ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  NN )
106105nnzd 10132 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )
10777nnne0d 9806 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
a ^ ( a 
pCnt  ( # `  B
) ) )  =/=  0 )
108 dvdscmulr 12573 . . . . . . . . . . . . . 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 ) ) ) ) ) )
10982, 106, 78, 107, 108syl112anc 1186 . . . . . . . . . . . . 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 ) ) ) ) ) )
110103, 109mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )
11117, 28odcl 14867 . . . . . . . . . . . . . . 15  |-  ( x  e.  B  ->  ( O `  x )  e.  NN0 )
112111adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( O `  x )  e.  NN0 )
113112nn0zd 10131 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  a  e.  A )  /\  p  e.  ( A  \  { a } ) )  /\  x  e.  B )  ->  ( O `  x )  e.  ZZ )
114 dvdstr 12578 . . . . . . . . . . . . 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
) ) ) ) ) )
115113, 82, 106, 114syl3anc 1182 . . . . . . . . . . . 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 ) ) ) ) ) )
116110, 115mpan2d 655 . . . . . . . . . . 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 )
) ) ) ) )
117116ss2rabdv 3267 . . . . . . . . . 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 )
) ) ) } )
11849elpw 3644 . . . . . . . . . 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 )
) ) ) } )
119117, 118sylibr 203 . . . . . . . . 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 )
) ) ) } )
12031reseq1i 4967 . . . . . . . . . 10  |-  ( S  |`  ( A  \  {
a } ) )  =  ( ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  |`  ( A  \  { a } ) )
121 difss 3316 . . . . . . . . . . 11  |-  ( A 
\  { a } )  C_  A
122 resmpt 5016 . . . . . . . . . . 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
) ) ) } ) )
123121, 122ax-mp 8 . . . . . . . . . 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 )
) ) } )
124120, 123eqtri 2316 . . . . . . . . 9  |-  ( S  |`  ( A  \  {
a } ) )  =  ( p  e.  ( A  \  {
a } )  |->  { x  e.  B  | 
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) ) } )
125119, 124fmptd 5700 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( S  |`  ( A  \  { a } ) ) : ( A 
\  { a } ) --> ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
126 frn 5411 . . . . . . . 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 )
) ) ) } )
127125, 126syl 15 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ran  ( S  |`  ( A 
\  { a } ) )  C_  ~P { x  e.  B  |  ( O `  x )  ||  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) ) } )
12860, 127syl5eqss 3235 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( S " ( A  \  { a } ) )  C_  ~P { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
129 sspwuni 4003 . . . . . 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 )
) ) ) } )
130128, 129sylib 188 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  U. ( S " ( A  \  { a } ) )  C_  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) } )
131104nnzd 10132 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )
13228, 17oddvdssubg 15163 . . . . . 6  |-  ( ( G  e.  Abel  /\  (
( # `  B )  /  ( a ^
( a  pCnt  ( # `
 B ) ) ) )  e.  ZZ )  ->  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  e.  (SubGrp `  G
) )
13354, 131, 132syl2anc 642 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  e.  (SubGrp `  G
) )
1343mrcsscl 13538 . . . . 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
) ) ) ) } )
13559, 130, 133, 134syl3anc 1182 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( A 
\  { a } ) ) )  C_  { x  e.  B  | 
( O `  x
)  ||  ( ( # `
 B )  / 
( a ^ (
a  pCnt  ( # `  B
) ) ) ) } )
136 ss2in 3409 . . . 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 )
) ) ) } ) )
13753, 135, 136syl2anc 642 . . 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 )
) ) ) } ) )
138 eqid 2296 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( a ^ (
a  pCnt  ( # `  B
) ) ) }
139 eqid 2296 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) }
14074simp2d 968 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
( a ^ (
a  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( a ^ ( a  pCnt  (
# `  B )
) ) ) )  =  1 )
141 eqid 2296 . . . . 5  |-  ( LSSum `  G )  =  (
LSSum `  G )
14217, 28, 138, 139, 54, 76, 104, 140, 101, 2, 141ablfacrp 15317 . . . 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 ) )
143142simpld 445 . . 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 ) } )
144137, 143sseqtrd 3227 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( S `  a
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( A  \  { a } ) ) ) )  C_  { ( 0g `  G
) } )
1451, 2, 3, 6, 13, 32, 41, 144dmdprdd 15253 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353    || cdivides 12547    gcd cgcd 12701   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   odcod 14856   LSSumclsm 14961   Abelcabel 15106   DProd cdprd 15247
This theorem is referenced by:  ablfac1c  15322  ablfac1eu  15324  ablfaclem2  15337  ablfaclem3  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-cntz 14809  df-od 14860  df-lsm 14963  df-cmn 15107  df-abl 15108  df-dprd 15249
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