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Theorem ablfac2 17800
Description: Choose generators for each cyclic group in ablfac 17799. (Contributed by Mario Carneiro, 28-Apr-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 )
ablfac2.m  |-  .x.  =  (.g
`  G )
ablfac2.s  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
Assertion
Ref Expression
ablfac2  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Distinct variable groups:    S, r    k, n, r, w, B    .x. , k, w    C, k, n, w    ph, k, n, w    k, G, n, r, w
Allowed substitution hints:    ph( r)    C( r)    S( w, k, n)    .x. ( n, r)

Proof of Theorem ablfac2
Dummy variables  s  x  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdf 12723 . . . . . . . 8  |-  ( s  e. Word  C  ->  s : ( 0..^ (
# `  s )
) --> C )
21ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : ( 0..^ (
# `  s )
) --> C )
3 fdm 5745 . . . . . . 7  |-  ( s : ( 0..^ (
# `  s )
) --> C  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
42, 3syl 17 . . . . . 6  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
5 fzofi 12225 . . . . . 6  |-  ( 0..^ ( # `  s
) )  e.  Fin
64, 5syl6eqel 2557 . . . . 5  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  e.  Fin )
74feq2d 5725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
s : dom  s --> C 
<->  s : ( 0..^ ( # `  s
) ) --> C ) )
82, 7mpbird 240 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : dom  s --> C )
98ffvelrnda 6037 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  C
)
10 oveq2 6316 . . . . . . . . . . . 12  |-  ( r  =  ( s `  k )  ->  ( Gs  r )  =  ( Gs  ( s `  k
) ) )
1110eleq1d 2533 . . . . . . . . . . 11  |-  ( r  =  ( s `  k )  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  (
s `  k )
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
12 ablfac.c . . . . . . . . . . 11  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
1311, 12elrab2 3186 . . . . . . . . . 10  |-  ( ( s `  k )  e.  C  <->  ( (
s `  k )  e.  (SubGrp `  G )  /\  ( Gs  ( s `  k ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
1413simplbi 467 . . . . . . . . 9  |-  ( ( s `  k )  e.  C  ->  (
s `  k )  e.  (SubGrp `  G )
)
159, 14syl 17 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  (SubGrp `  G ) )
16 ablfac.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
1716subgss 16896 . . . . . . . 8  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  C_  B
)
1815, 17syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  C_  B
)
19 inss1 3643 . . . . . . . . . . 11  |-  (CycGrp  i^i  ran pGrp  )  C_ CycGrp
2013simprbi 471 . . . . . . . . . . . 12  |-  ( ( s `  k )  e.  C  ->  ( Gs  ( s `  k
) )  e.  (CycGrp 
i^i  ran pGrp  ) )
219, 20syl 17 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e.  (CycGrp  i^i  ran pGrp  ) )
2219, 21sseldi 3416 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e. CycGrp )
23 eqid 2471 . . . . . . . . . . . 12  |-  ( Base `  ( Gs  ( s `  k ) ) )  =  ( Base `  ( Gs  ( s `  k
) ) )
24 eqid 2471 . . . . . . . . . . . 12  |-  (.g `  ( Gs  ( s `  k
) ) )  =  (.g `  ( Gs  ( s `
 k ) ) )
2523, 24iscyg 17592 . . . . . . . . . . 11  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  <->  ( ( Gs  ( s `  k
) )  e.  Grp  /\ 
E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
2625simprbi 471 . . . . . . . . . 10  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  ->  E. x  e.  ( Base `  ( Gs  ( s `  k ) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
2722, 26syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
28 eqid 2471 . . . . . . . . . . . 12  |-  ( Gs  ( s `  k ) )  =  ( Gs  ( s `  k ) )
2928subgbas 16899 . . . . . . . . . . 11  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3015, 29syl 17 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3130rexeqdv 2980 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) )  <->  E. x  e.  ( Base `  ( Gs  ( s `  k
) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k
) ) ) x ) )  =  (
Base `  ( Gs  (
s `  k )
) ) ) )
3227, 31mpbird 240 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
3315ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( s `  k
)  e.  (SubGrp `  G ) )
34 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
35 simplr 770 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  x  e.  ( s `
 k ) )
36 ablfac2.m . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
3736, 28, 24subgmulg 16909 . . . . . . . . . . . . 13  |-  ( ( ( s `  k
)  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  x  e.  ( s `  k
) )  ->  (
n  .x.  x )  =  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )
3833, 34, 35, 37syl3anc 1292 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )
3938mpteq2dva 4482 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4039rneqd 5068 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4130adantr 472 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( s `  k
)  =  ( Base `  ( Gs  ( s `  k ) ) ) )
4240, 41eqeq12d 2486 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4342rexbidva 2889 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4432, 43mpbird 240 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
) )
45 ssrexv 3480 . . . . . . 7  |-  ( ( s `  k ) 
C_  B  ->  ( E. x  e.  (
s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) ) )
4618, 44, 45sylc 61 . . . . . 6  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )
4746ralrimiva 2809 . . . . 5  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  ( s `  k ) )
48 oveq2 6316 . . . . . . . . 9  |-  ( x  =  ( w `  k )  ->  (
n  .x.  x )  =  ( n  .x.  ( w `  k
) ) )
4948mpteq2dv 4483 . . . . . . . 8  |-  ( x  =  ( w `  k )  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) ) )
5049rneqd 5068 . . . . . . 7  |-  ( x  =  ( w `  k )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )
5150eqeq1d 2473 . . . . . 6  |-  ( x  =  ( w `  k )  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) )  =  ( s `  k ) ) )
5251ac6sfi 7833 . . . . 5  |-  ( ( dom  s  e.  Fin  /\ 
A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )  ->  E. w ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
536, 47, 52syl2anc 673 . . . 4  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
54 simprl 772 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w : dom  s --> B )
554adantr 472 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  s  =  ( 0..^ ( # `  s ) ) )
5655feq2d 5725 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w : dom  s --> B  <->  w :
( 0..^ ( # `  s ) ) --> B ) )
5754, 56mpbid 215 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w :
( 0..^ ( # `  s ) ) --> B )
58 iswrdi 12722 . . . . . . . 8  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  w  e. Word  B )
5957, 58syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w  e. Word  B )
60 fdm 5745 . . . . . . . . . . . . . 14  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  dom  w  =  ( 0..^ ( # `  s
) ) )
6157, 60syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  ( 0..^ ( # `  s ) ) )
6261, 55eqtr4d 2508 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  dom  s )
6362eleq2d 2534 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( j  e.  dom  w  <->  j  e.  dom  s ) )
6463biimpa 492 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  -> 
j  e.  dom  s
)
65 simprr 774 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
66 simpl 464 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  k  =  j )
6766fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( w `  k
)  =  ( w `
 j ) )
6867oveq2d 6324 . . . . . . . . . . . . . . . 16  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( n  .x.  (
w `  k )
)  =  ( n 
.x.  ( w `  j ) ) )
6968mpteq2dva 4482 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
7069rneqd 5068 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
71 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
s `  k )  =  ( s `  j ) )
7270, 71eqeq12d 2486 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) )  =  ( s `  j ) ) )
7372rspccva 3135 . . . . . . . . . . . 12  |-  ( ( A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) )  =  ( s `  k
)  /\  j  e.  dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7465, 73sylan 479 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
758adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> C )
7675ffvelrnda 6037 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  -> 
( s `  j
)  e.  C )
7774, 76eqeltrd 2549 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
7864, 77syldan 478 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
79 ablfac2.s . . . . . . . . . 10  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
80 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
w `  k )  =  ( w `  j ) )
8180oveq2d 6324 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  (
n  .x.  ( w `  k ) )  =  ( n  .x.  (
w `  j )
) )
8281mpteq2dv 4483 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
8382rneqd 5068 . . . . . . . . . . 11  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
8483cbvmptv 4488 . . . . . . . . . 10  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8579, 84eqtri 2493 . . . . . . . . 9  |-  S  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8678, 85fmptd 6061 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S : dom  w --> C )
87 simprl 772 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  G dom DProd  s )
8887adantr 472 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  s )
8962raleqdv 2979 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( A. k  e.  dom  w ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
9065, 89mpbird 240 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
91 mpteq12 4475 . . . . . . . . . . . 12  |-  ( ( dom  w  =  dom  s  /\  A. k  e. 
dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9262, 90, 91syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( k  e.  dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9379, 92syl5eq 2517 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
94 dprdf 17716 . . . . . . . . . . . 12  |-  ( G dom DProd  s  ->  s : dom  s --> (SubGrp `  G ) )
9588, 94syl 17 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> (SubGrp `  G )
)
9695feqmptd 5932 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
9793, 96eqtr4d 2508 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  s )
9888, 97breqtrrd 4422 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  S )
9997oveq2d 6324 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  ( G DProd 
s ) )
100 simplrr 779 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  s )  =  B )
10199, 100eqtrd 2505 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  B )
10286, 98, 1013jca 1210 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
10359, 102jca 541 . . . . . 6  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) )
104103ex 441 . . . . 5  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) ) )
105104eximdv 1772 . . . 4  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  ( E. w ( w : dom  s --> B  /\  A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  E. w ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) ) )
10653, 105mpd 15 . . 3  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
107 df-rex 2762 . . 3  |-  ( E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B )  <->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
108106, 107sylibr 217 . 2  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) )
109 ablfac.1 . . 3  |-  ( ph  ->  G  e.  Abel )
110 ablfac.2 . . 3  |-  ( ph  ->  B  e.  Fin )
11116, 12, 109, 110ablfac 17799 . 2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
112108, 111r19.29a 2918 1  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760    i^i cin 3389    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   0cc0 9557   ZZcz 10961  ..^cfzo 11942   #chash 12553  Word cword 12703   Basecbs 15199   ↾s cress 15200   Grpcgrp 16747  .gcmg 16750  SubGrpcsubg 16889   pGrp cpgp 17247   Abelcabl 17509  CycGrpccyg 17590   DProd cdprd 17703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-rpss 6590  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-eqg 16894  df-ghm 16959  df-gim 17001  df-ga 17022  df-cntz 17049  df-oppg 17075  df-od 17250  df-gex 17252  df-pgp 17254  df-lsm 17366  df-pj1 17367  df-cmn 17510  df-abl 17511  df-cyg 17591  df-dprd 17705
This theorem is referenced by:  dchrpt  24274
  Copyright terms: Public domain W3C validator