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Theorem ablfaclem3 15572
Description: Lemma for ablfac 15573. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
ablfac.b  |-  B  =  ( Base `  G
)
ablfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
ablfac.1  |-  ( ph  ->  G  e.  Abel )
ablfac.2  |-  ( ph  ->  B  e.  Fin )
ablfac.o  |-  O  =  ( od `  G
)
ablfac.a  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
ablfac.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac.w  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
Assertion
Ref Expression
ablfaclem3  |-  ( ph  ->  ( W `  B
)  =/=  (/) )
Distinct variable groups:    s, p, x, A    g, r, s, S    g, p, w, x, B, r, s    O, p, x    C, g, p, s, w, x    W, p, w, x    ph, p, s, w, x    g, G, p, r, s, w, x
Allowed substitution hints:    ph( g, r)    A( w, g, r)    C( r)    S( x, w, p)    O( w, g, s, r)    W( g, s, r)

Proof of Theorem ablfaclem3
Dummy variables  a 
b  c  f  h  q  t  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 11239 . . . 4  |-  ( ph  ->  ( 1 ... ( # `
 B ) )  e.  Fin )
2 ablfac.a . . . . 5  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
3 prmnn 13009 . . . . . . . 8  |-  ( w  e.  Prime  ->  w  e.  NN )
433ad2ant2 979 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  NN )
5 prmz 13010 . . . . . . . . 9  |-  ( w  e.  Prime  ->  w  e.  ZZ )
6 ablfac.1 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 15344 . . . . . . . . . . 11  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac.b . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
98grpbn0 14761 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 19 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
11 ablfac.2 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 11572 . . . . . . . . . . 11  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 224 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  NN )
15 dvdsle 12822 . . . . . . . . 9  |-  ( ( w  e.  ZZ  /\  ( # `  B )  e.  NN )  -> 
( w  ||  ( # `
 B )  ->  w  <_  ( # `  B
) ) )
165, 14, 15syl2anr 465 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime )  ->  ( w  ||  ( # `  B
)  ->  w  <_  (
# `  B )
) )
17163impia 1150 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  <_  ( # `
 B ) )
1814nnzd 10306 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
19183ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( # `  B
)  e.  ZZ )
20 fznn 11045 . . . . . . . 8  |-  ( (
# `  B )  e.  ZZ  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
2119, 20syl 16 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
224, 17, 21mpbir2and 889 . . . . . 6  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  ( 1 ... ( # `  B ) ) )
2322rabssdv 3366 . . . . 5  |-  ( ph  ->  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  ( 1 ... ( # `
 B ) ) )
242, 23syl5eqss 3335 . . . 4  |-  ( ph  ->  A  C_  ( 1 ... ( # `  B
) ) )
25 ssfi 7265 . . . 4  |-  ( ( ( 1 ... ( # `
 B ) )  e.  Fin  /\  A  C_  ( 1 ... ( # `
 B ) ) )  ->  A  e.  Fin )
261, 24, 25syl2anc 643 . . 3  |-  ( ph  ->  A  e.  Fin )
27 dfin5 3271 . . . . . . . 8  |-  (Word  C  i^i  ( W `  ( S `  q )
) )  =  {
y  e. Word  C  | 
y  e.  ( W `
 ( S `  q ) ) }
28 ablfac.o . . . . . . . . . . . . . 14  |-  O  =  ( od `  G
)
29 ablfac.s . . . . . . . . . . . . . 14  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
30 ssrab2 3371 . . . . . . . . . . . . . . . 16  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  Prime
312, 30eqsstri 3321 . . . . . . . . . . . . . . 15  |-  A  C_  Prime
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  C_  Prime )
338, 28, 29, 6, 11, 32ablfac1b 15555 . . . . . . . . . . . . 13  |-  ( ph  ->  G dom DProd  S )
34 fvex 5682 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
358, 34eqeltri 2457 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
3635rabex 4295 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3736, 29dmmpti 5514 . . . . . . . . . . . . . 14  |-  dom  S  =  A
3837a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  S  =  A )
3933, 38dprdf2 15492 . . . . . . . . . . . 12  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
4039ffvelrnda 5809 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
41 ablfac.c . . . . . . . . . . . 12  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
42 ablfac.w . . . . . . . . . . . 12  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
438, 41, 6, 11, 28, 2, 29, 42ablfaclem1 15570 . . . . . . . . . . 11  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( W `  ( S `  q
) )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } )
4440, 43syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) } )
45 ssrab2 3371 . . . . . . . . . 10  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } 
C_ Word  C
4644, 45syl6eqss 3341 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  C_ Word  C )
47 dfss1 3488 . . . . . . . . 9  |-  ( ( W `  ( S `
 q ) ) 
C_ Word  C  <->  (Word  C  i^i  ( W `  ( S `
 q ) ) )  =  ( W `
 ( S `  q ) ) )
4846, 47sylib 189 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  (Word  C  i^i  ( W `  ( S `  q ) ) )  =  ( W `  ( S `
 q ) ) )
4927, 48syl5eqr 2433 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =  ( W `  ( S `
 q ) ) )
5049, 44eqtrd 2419 . . . . . 6  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } )
51 eqid 2387 . . . . . . . . 9  |-  ( Base `  ( Gs  ( S `  q ) ) )  =  ( Base `  ( Gs  ( S `  q ) ) )
52 eqid 2387 . . . . . . . . 9  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  =  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }
536adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  G  e.  Abel )
54 eqid 2387 . . . . . . . . . . 11  |-  ( Gs  ( S `  q ) )  =  ( Gs  ( S `  q ) )
5554subgabl 15382 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  ( S `  q )  e.  (SubGrp `  G )
)  ->  ( Gs  ( S `  q )
)  e.  Abel )
5653, 40, 55syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Abel )
5732sselda 3291 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
5854subgbas 14875 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  =  (
Base `  ( Gs  ( S `  q )
) ) )
5940, 58syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  =  ( Base `  ( Gs  ( S `  q ) ) ) )
6059fveq2d 5672 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )
618, 28, 29, 6, 11, 32ablfac1a 15554 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6260, 61eqtr3d 2421 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6362oveq2d 6036 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  (
q ^ ( q 
pCnt  ( # `  B
) ) ) ) )
6414adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 B )  e.  NN )
6557, 64pccld 13151 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  NN0 )
6665nn0zd 10305 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  ZZ )
67 pcid 13173 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Prime  /\  (
q  pCnt  ( # `  B
) )  e.  ZZ )  ->  ( q  pCnt  ( q ^ ( q 
pCnt  ( # `  B
) ) ) )  =  ( q  pCnt  (
# `  B )
) )
6857, 66, 67syl2anc 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( q ^ ( q  pCnt  (
# `  B )
) ) )  =  ( q  pCnt  ( # `
 B ) ) )
6963, 68eqtrd 2419 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  ( # `
 B ) ) )
7069oveq2d 6036 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) )  =  ( q ^
( q  pCnt  ( # `
 B ) ) ) )
7162, 70eqtr4d 2422 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) )
7254subggrp 14874 . . . . . . . . . . . 12  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( Gs  ( S `  q )
)  e.  Grp )
7340, 72syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Grp )
7411adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  B  e.  Fin )
758subgss 14872 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  C_  B
)
7640, 75syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  C_  B )
77 ssfi 7265 . . . . . . . . . . . . 13  |-  ( ( B  e.  Fin  /\  ( S `  q ) 
C_  B )  -> 
( S `  q
)  e.  Fin )
7874, 76, 77syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  Fin )
7959, 78eqeltrrd 2462 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( S `
 q ) ) )  e.  Fin )
8051pgpfi2 15167 . . . . . . . . . . 11  |-  ( ( ( Gs  ( S `  q ) )  e. 
Grp  /\  ( Base `  ( Gs  ( S `  q ) ) )  e.  Fin )  -> 
( q pGrp  ( Gs  ( S `  q ) )  <->  ( q  e. 
Prime  /\  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) )  =  ( q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) ) ) )
8173, 79, 80syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q pGrp  ( Gs  ( S `
 q ) )  <-> 
( q  e.  Prime  /\  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) )  =  ( q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) ) ) )
8257, 71, 81mpbir2and 889 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  q pGrp  ( Gs  ( S `  q ) ) )
8351, 52, 56, 82, 79pgpfac 15569 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  E. s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  (
( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) ) ) )
84 ssrab2 3371 . . . . . . . . . . . . . 14  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  ( Gs  ( S `  q ) ) )
85 sswrd 11664 . . . . . . . . . . . . . 14  |-  ( { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_  (SubGrp `  ( Gs  ( S `
 q ) ) )  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q )
) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )
8684, 85ax-mp 8 . . . . . . . . . . . . 13  |- Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  (SubGrp `  ( Gs  ( S `  q ) ) )
8786sseli 3287 . . . . . . . . . . . 12  |-  ( s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  ->  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )
8840adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( S `  q
)  e.  (SubGrp `  G ) )
8988adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  e.  (SubGrp `  G ) )
9054subgdmdprd 15519 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( ( Gs  ( S `  q ) ) dom DProd  s  <->  ( G dom DProd  s  /\  ran  s  C_ 
~P ( S `  q ) ) ) )
9188, 90syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( ( Gs  ( S `
 q ) ) dom DProd  s  <->  ( G dom DProd  s  /\  ran  s  C_ 
~P ( S `  q ) ) ) )
9291simprbda 607 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  G dom DProd  s )
9391simplbda 608 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ran  s  C_  ~P ( S `  q ) )
9454, 89, 92, 93subgdprd 15520 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  ( G DProd  s ) )
9559ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  =  ( Base `  ( Gs  ( S `  q ) ) ) )
9695eqcomd 2392 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( Base `  ( Gs  ( S `  q ) ) )  =  ( S `  q ) )
9794, 96eqeq12d 2401 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  <->  ( G DProd  s )  =  ( S `
 q ) ) )
9897biimpd 199 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  ->  ( G DProd  s )  =  ( S `  q ) ) )
9998, 92jctild 528 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  ->  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
10099expimpd 587 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( ( ( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
10187, 100sylan2 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  ( ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
102 oveq2 6028 . . . . . . . . . . . . . . . 16  |-  ( r  =  y  ->  (
( Gs  ( S `  q ) )s  r )  =  ( ( Gs  ( S `  q ) )s  y ) )
103102eleq1d 2453 . . . . . . . . . . . . . . 15  |-  ( r  =  y  ->  (
( ( Gs  ( S `
 q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) 
<->  ( ( Gs  ( S `
 q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) ) )
104103cbvrabv 2898 . . . . . . . . . . . . . 14  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  =  { y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) }
10554subsubg 14890 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  <->  ( y  e.  (SubGrp `  G )  /\  y  C_  ( S `
 q ) ) ) )
10640, 105syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  q  e.  A )  ->  (
y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  <-> 
( y  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) ) ) )
107106simprbda 607 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  e.  (SubGrp `  G ) )
1081073adant3 977 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  e.  (SubGrp `  G ) )
109403ad2ant1 978 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( S `  q )  e.  (SubGrp `  G ) )
110106simplbda 608 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  C_  ( S `  q ) )
1111103adant3 977 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  C_  ( S `  q ) )
112 ressabs 13454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S `  q
)  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) )  ->  (
( Gs  ( S `  q ) )s  y )  =  ( Gs  y ) )
113109, 111, 112syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( ( Gs  ( S `  q ) )s  y )  =  ( Gs  y ) )
114 simp3 959 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp 
i^i  ran pGrp  ) )
115113, 114eqeltrrd 2462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( Gs  y
)  e.  (CycGrp  i^i  ran pGrp  ) )
116 oveq2 6028 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  y  ->  ( Gs  r )  =  ( Gs  y ) )
117116eleq1d 2453 . . . . . . . . . . . . . . . . 17  |-  ( r  =  y  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  y
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
118117, 41elrab2 3037 . . . . . . . . . . . . . . . 16  |-  ( y  e.  C  <->  ( y  e.  (SubGrp `  G )  /\  ( Gs  y )  e.  (CycGrp  i^i  ran pGrp  ) ) )
119108, 115, 118sylanbrc 646 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  e.  C )
120119rabssdv 3366 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  { y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
121104, 120syl5eqss 3335 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
122 sswrd 11664 . . . . . . . . . . . . 13  |-  ( { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_  C  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_ Word  C )
123121, 122syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  C )
124123sselda 3291 . . . . . . . . . . 11  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  s  e. Word  C
)
125101, 124jctild 528 . . . . . . . . . 10  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  ( ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( s  e. Word  C  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) ) )
126125expimpd 587 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  (
( s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  /\  ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) ) )  ->  ( s  e. Word  C  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) ) ) )
127126reximdv2 2758 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  ( E. s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  ( ( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
12883, 127mpd 15 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) )
129 rabn0 3590 . . . . . . 7  |-  ( { s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) }  =/=  (/)  <->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) )
130128, 129sylibr 204 . . . . . 6  |-  ( (
ph  /\  q  e.  A )  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) }  =/=  (/) )
13150, 130eqnetrd 2568 . . . . 5  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =/=  (/) )
132 rabn0 3590 . . . . 5  |-  ( { y  e. Word  C  | 
y  e.  ( W `
 ( S `  q ) ) }  =/=  (/)  <->  E. y  e. Word  C
y  e.  ( W `
 ( S `  q ) ) )
133131, 132sylib 189 . . . 4  |-  ( (
ph  /\  q  e.  A )  ->  E. y  e. Word  C y  e.  ( W `  ( S `
 q ) ) )
134133ralrimiva 2732 . . 3  |-  ( ph  ->  A. q  e.  A  E. y  e. Word  C y  e.  ( W `  ( S `  q ) ) )
135 eleq1 2447 . . . 4  |-  ( y  =  ( f `  q )  ->  (
y  e.  ( W `
 ( S `  q ) )  <->  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )
136135ac6sfi 7287 . . 3  |-  ( ( A  e.  Fin  /\  A. q  e.  A  E. y  e. Word  C y  e.  ( W `  ( S `  q )
) )  ->  E. f
( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )
13726, 134, 136syl2anc 643 . 2  |-  ( ph  ->  E. f ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )
138 sneq 3768 . . . . . . . . 9  |-  ( q  =  y  ->  { q }  =  { y } )
139 fveq2 5668 . . . . . . . . . 10  |-  ( q  =  y  ->  (
f `  q )  =  ( f `  y ) )
140139dmeqd 5012 . . . . . . . . 9  |-  ( q  =  y  ->  dom  ( f `  q
)  =  dom  (
f `  y )
)
141138, 140xpeq12d 4843 . . . . . . . 8  |-  ( q  =  y  ->  ( { q }  X.  dom  ( f `  q
) )  =  ( { y }  X.  dom  ( f `  y
) ) )
142141cbviunv 4071 . . . . . . 7  |-  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) )  =  U_ y  e.  A  ( { y }  X.  dom  ( f `  y
) )
14326adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  A  e.  Fin )
144 snfi 7123 . . . . . . . . . 10  |-  { y }  e.  Fin
145 simprl 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  f : A -->Word  C )
146145ffvelrnda 5809 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  ( f `  y
)  e. Word  C )
147 wrdf 11660 . . . . . . . . . . . 12  |-  ( ( f `  y )  e. Word  C  ->  (
f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C )
148 fdm 5535 . . . . . . . . . . . 12  |-  ( ( f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C  ->  dom  ( f `
 y )  =  ( 0..^ ( # `  ( f `  y
) ) ) )
149146, 147, 1483syl 19 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  =  ( 0..^ ( # `  (
f `  y )
) ) )
150 fzofi 11240 . . . . . . . . . . 11  |-  ( 0..^ ( # `  (
f `  y )
) )  e.  Fin
151149, 150syl6eqel 2475 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  e.  Fin )
152 xpfi 7314 . . . . . . . . . 10  |-  ( ( { y }  e.  Fin  /\  dom  ( f `
 y )  e. 
Fin )  ->  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )
153144, 151, 152sylancr 645 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  ( { y }  X.  dom  ( f `
 y ) )  e.  Fin )
154153ralrimiva 2732 . . . . . . . 8  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  A. y  e.  A  ( {
y }  X.  dom  ( f `  y
) )  e.  Fin )
155 iunfi 7330 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  A. y  e.  A  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )  ->  U_ y  e.  A  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )
156143, 154, 155syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  U_ y  e.  A  ( {
y }  X.  dom  ( f `  y
) )  e.  Fin )
157142, 156syl5eqel 2471 . . . . . 6  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) )  e.  Fin )
158 hashcl 11566 . . . . . 6  |-  ( U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  e.  Fin  ->  ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )  e. 
NN0 )
159 hashfzo0 11622 . . . . . 6  |-  ( (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) )  e.  NN0  ->  ( # `  (
0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
160157, 158, 1593syl 19 . . . . 5  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( # `
 ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) )  =  ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )
161 fzofi 11240 . . . . . 6  |-  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  e.  Fin
162 hashen 11558 . . . . . 6  |-  ( ( ( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )  e.  Fin  /\  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  e.  Fin )  ->  ( ( # `  ( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )  <->  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
163161, 157, 162sylancr 645 . . . . 5  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
( # `  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )  <->  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
164160, 163mpbid 202 . . . 4  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )
165 bren 7053 . . . 4  |-  ( ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  <->  E. h  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )
166164, 165sylib 189 . . 3  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  E. h  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )
1676adantr 452 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  G  e.  Abel )
16811adantr 452 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  B  e.  Fin )
169 breq1 4156 . . . . . . . 8  |-  ( w  =  a  ->  (
w  ||  ( # `  B
)  <->  a  ||  ( # `
 B ) ) )
170169cbvrabv 2898 . . . . . . 7  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  =  { a  e.  Prime  |  a  ||  ( # `  B
) }
1712, 170eqtri 2407 . . . . . 6  |-  A  =  { a  e.  Prime  |  a  ||  ( # `  B ) }
172 fveq2 5668 . . . . . . . . . . 11  |-  ( x  =  c  ->  ( O `  x )  =  ( O `  c ) )
173172breq1d 4163 . . . . . . . . . 10  |-  ( x  =  c  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) ) )
174173cbvrabv 2898 . . . . . . . . 9  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }
175 id 20 . . . . . . . . . . . 12  |-  ( p  =  b  ->  p  =  b )
176 oveq1 6027 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
p  pCnt  ( # `  B
) )  =  ( b  pCnt  ( # `  B
) ) )
177175, 176oveq12d 6038 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( b ^ (
b  pCnt  ( # `  B
) ) ) )
178177breq2d 4165 . . . . . . . . . 10  |-  ( p  =  b  ->  (
( O `  c
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) ) )
179178rabbidv 2891 . . . . . . . . 9  |-  ( p  =  b  ->  { c  e.  B  |  ( O `  c ) 
||  ( p ^
( p  pCnt  ( # `
 B ) ) ) }  =  {
c  e.  B  | 
( O `  c
)  ||  ( b ^ ( b  pCnt  (
# `  B )
) ) } )
180174, 179syl5eq 2431 . . . . . . . 8  |-  ( p  =  b  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( b ^ (
b  pCnt  ( # `  B
) ) ) } )
181180cbvmptv 4241 . . . . . . 7  |-  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c ) 
||  ( b ^
( b  pCnt  ( # `
 B ) ) ) } )
18229, 181eqtri 2407 . . . . . 6  |-  S  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) } )
183 breq2 4157 . . . . . . . . . 10  |-  ( s  =  t  ->  ( G dom DProd  s  <->  G dom DProd  t ) )
184 oveq2 6028 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( G DProd  s )  =  ( G DProd  t ) )
185184eqeq1d 2395 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( G DProd  s )  =  g  <->  ( G DProd  t
)  =  g ) )
186183, 185anbi12d 692 . . . . . . . . 9  |-  ( s  =  t  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) ) )
187186cbvrabv 2898 . . . . . . . 8  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) }
188187mpteq2i 4233 . . . . . . 7  |-  ( g  e.  (SubGrp `  G
)  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
s )  =  g ) } )  =  ( g  e.  (SubGrp `  G )  |->  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd  t )  =  g ) } )
18942, 188eqtri 2407 . . . . . 6  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd  t )  =  g ) } )
190 simprll 739 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  f : A -->Word  C )
191 simprlr 740 . . . . . . 7  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  A. q  e.  A  ( f `  q
)  e.  ( W `
 ( S `  q ) ) )
192 fveq2 5668 . . . . . . . . . 10  |-  ( q  =  y  ->  ( S `  q )  =  ( S `  y ) )
193192fveq2d 5672 . . . . . . . . 9  |-  ( q  =  y  ->  ( W `  ( S `  q ) )  =  ( W `  ( S `  y )
) )
194139, 193eleq12d 2455 . . . . . . . 8  |-  ( q  =  y  ->  (
( f `  q
)  e.  ( W `
 ( S `  q ) )  <->  ( f `  y )  e.  ( W `  ( S `
 y ) ) ) )
195194cbvralv 2875 . . . . . . 7  |-  ( A. q  e.  A  (
f `  q )  e.  ( W `  ( S `  q )
)  <->  A. y  e.  A  ( f `  y
)  e.  ( W `
 ( S `  y ) ) )
196191, 195sylib 189 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  A. y  e.  A  ( f `  y
)  e.  ( W `
 ( S `  y ) ) )
197 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  h : ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )
1988, 41, 167, 168, 28, 171, 182, 189, 190, 196, 142, 197ablfaclem2 15571 . . . . 5  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  ( W `  B )  =/=  (/) )
199198expr 599 . . . 4  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
h : ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  ->  ( W `  B )  =/=  (/) ) )
200199exlimdv 1643 . . 3  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( E. h  h :
( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  ->  ( W `  B )  =/=  (/) ) )
201166, 200mpd 15 . 2  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( W `  B )  =/=  (/) )
202137, 201exlimddv 1645 1  |-  ( ph  ->  ( W `  B
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653   _Vcvv 2899    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   U_ciun 4035   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   dom cdm 4818   ran crn 4819   -->wf 5390   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    ~~ cen 7042   Fincfn 7045   0cc0 8923   1c1 8924    <_ cle 9054   NNcn 9932   NN0cn0 10153   ZZcz 10214   ...cfz 10975  ..^cfzo 11065   ^cexp 11309   #chash 11545  Word cword 11644    || cdivides 12779   Primecprime 13006    pCnt cpc 13137   Basecbs 13396   ↾s cress 13397   Grpcgrp 14612  SubGrpcsubg 14865   odcod 15090   pGrp cpgp 15092   Abelcabel 15340  CycGrpccyg 15414   DProd cdprd 15481
This theorem is referenced by:  ablfac  15573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-rpss 6458  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6841  df-ec 6843  df-qs 6847  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-word 11650  df-concat 11651  df-s1 11652  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-eqg 14870  df-ghm 14931  df-gim 14973  df-ga 14994  df-cntz 15043  df-oppg 15069  df-od 15094  df-gex 15095  df-pgp 15096  df-lsm 15197  df-pj1 15198  df-cmn 15341  df-abl 15342  df-cyg 15415  df-dprd 15483
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