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Theorem ablfac2 16942
Description: Choose generators for each cyclic group in ablfac 16941. (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 ablfac.b . . 3  |-  B  =  ( Base `  G
)
2 ablfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
3 ablfac.1 . . 3  |-  ( ph  ->  G  e.  Abel )
4 ablfac.2 . . 3  |-  ( ph  ->  B  e.  Fin )
51, 2, 3, 4ablfac 16941 . 2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
6 wrdf 12519 . . . . . . . . . 10  |-  ( s  e. Word  C  ->  s : ( 0..^ (
# `  s )
) --> C )
76ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : ( 0..^ (
# `  s )
) --> C )
8 fdm 5735 . . . . . . . . 9  |-  ( s : ( 0..^ (
# `  s )
) --> C  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
97, 8syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
10 fzofi 12052 . . . . . . . 8  |-  ( 0..^ ( # `  s
) )  e.  Fin
119, 10syl6eqel 2563 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  e.  Fin )
129feq2d 5718 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
s : dom  s --> C 
<->  s : ( 0..^ ( # `  s
) ) --> C ) )
137, 12mpbird 232 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : dom  s --> C )
1413ffvelrnda 6021 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  C
)
15 oveq2 6292 . . . . . . . . . . . . . 14  |-  ( r  =  ( s `  k )  ->  ( Gs  r )  =  ( Gs  ( s `  k
) ) )
1615eleq1d 2536 . . . . . . . . . . . . 13  |-  ( r  =  ( s `  k )  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  (
s `  k )
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
1716, 2elrab2 3263 . . . . . . . . . . . 12  |-  ( ( s `  k )  e.  C  <->  ( (
s `  k )  e.  (SubGrp `  G )  /\  ( Gs  ( s `  k ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
1817simplbi 460 . . . . . . . . . . 11  |-  ( ( s `  k )  e.  C  ->  (
s `  k )  e.  (SubGrp `  G )
)
1914, 18syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  (SubGrp `  G ) )
201subgss 16007 . . . . . . . . . 10  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  C_  B
)
2119, 20syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  C_  B
)
22 inss1 3718 . . . . . . . . . . . . 13  |-  (CycGrp  i^i  ran pGrp  )  C_ CycGrp
2317simprbi 464 . . . . . . . . . . . . . 14  |-  ( ( s `  k )  e.  C  ->  ( Gs  ( s `  k
) )  e.  (CycGrp 
i^i  ran pGrp  ) )
2414, 23syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( 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  ) )
2522, 24sseldi 3502 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e. CycGrp )
26 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( s `  k ) ) )  =  ( Base `  ( Gs  ( s `  k
) ) )
27 eqid 2467 . . . . . . . . . . . . . 14  |-  (.g `  ( Gs  ( s `  k
) ) )  =  (.g `  ( Gs  ( s `
 k ) ) )
2826, 27iscyg 16685 . . . . . . . . . . . . 13  |-  ( ( 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
) ) ) ) )
2928simprbi 464 . . . . . . . . . . . 12  |-  ( ( 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
) ) ) )
3025, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( ( 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
) ) ) )
31 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Gs  ( s `  k ) )  =  ( Gs  ( s `  k ) )
3231subgbas 16010 . . . . . . . . . . . . 13  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3319, 32syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3433rexeqdv 3065 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
) ) ) )
3530, 34mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( 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
) ) ) )
3619ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
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 ) )
37 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
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 )
38 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
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 ) )
39 ablfac2.m . . . . . . . . . . . . . . . 16  |-  .x.  =  (.g
`  G )
4039, 31, 27subgmulg 16020 . . . . . . . . . . . . . . 15  |-  ( ( ( s `  k
)  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  x  e.  ( s `  k
) )  ->  (
n  .x.  x )  =  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )
4136, 37, 38, 40syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
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 ) )
4241mpteq2dva 4533 . . . . . . . . . . . . 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  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4342rneqd 5230 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 ) ) )
4433adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 ) ) ) )
4543, 44eqeq12d 2489 . . . . . . . . . . 11  |-  ( ( ( ( ( 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
) ) ) ) )
4645rexbidva 2970 . . . . . . . . . 10  |-  ( ( ( ( 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
) ) ) ) )
4735, 46mpbird 232 . . . . . . . . 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  .x.  x ) )  =  ( s `  k
) )
48 ssrexv 3565 . . . . . . . . 9  |-  ( ( 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 ) ) )
4921, 47, 48sylc 60 . . . . . . . 8  |-  ( ( ( ( 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 ) )
5049ralrimiva 2878 . . . . . . 7  |-  ( ( ( 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 ) )
51 oveq2 6292 . . . . . . . . . . 11  |-  ( x  =  ( w `  k )  ->  (
n  .x.  x )  =  ( n  .x.  ( w `  k
) ) )
5251mpteq2dv 4534 . . . . . . . . . 10  |-  ( x  =  ( w `  k )  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) ) )
5352rneqd 5230 . . . . . . . . 9  |-  ( x  =  ( w `  k )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )
5453eqeq1d 2469 . . . . . . . 8  |-  ( x  =  ( w `  k )  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) )  =  ( s `  k ) ) )
5554ac6sfi 7764 . . . . . . 7  |-  ( ( 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 ) ) )
5611, 50, 55syl2anc 661 . . . . . 6  |-  ( ( ( 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 ) ) )
57 simprl 755 . . . . . . . . . . 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 ) ) )  ->  w : dom  s --> B )
589adantr 465 . . . . . . . . . . . 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  s  =  ( 0..^ ( # `  s ) ) )
5958feq2d 5718 . . . . . . . . . . 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 ) ) )  ->  ( w : dom  s --> B  <->  w :
( 0..^ ( # `  s ) ) --> B ) )
6057, 59mpbid 210 . . . . . . . . . 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 ) ) )  ->  w :
( 0..^ ( # `  s ) ) --> B )
61 iswrdi 12518 . . . . . . . . . 10  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  w  e. Word  B )
6260, 61syl 16 . . . . . . . . 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  e. Word  B )
63 fdm 5735 . . . . . . . . . . . . . . . 16  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  dom  w  =  ( 0..^ ( # `  s
) ) )
6460, 63syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 ) ) )
6564, 58eqtr4d 2511 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )
6665eleq2d 2537 . . . . . . . . . . . . 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 ) ) )  ->  ( j  e.  dom  w  <->  j  e.  dom  s ) )
6766biimpa 484 . . . . . . . . . . . 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 ) ) )  /\  j  e. 
dom  w )  -> 
j  e.  dom  s
)
68 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 ) )
69 simpl 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  k  =  j )
7069fveq2d 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( w `  k
)  =  ( w `
 j ) )
7170oveq2d 6300 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( n  .x.  (
w `  k )
)  =  ( n 
.x.  ( w `  j ) ) )
7271mpteq2dva 4533 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
7372rneqd 5230 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
74 fveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
s `  k )  =  ( s `  j ) )
7573, 74eqeq12d 2489 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) )  =  ( s `  j ) ) )
7675rspccva 3213 . . . . . . . . . . . . . 14  |-  ( ( 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 ) )
7768, 76sylan 471 . . . . . . . . . . . . 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 ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7813adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )
7978ffvelrnda 6021 . . . . . . . . . . . . 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 ) ) )  /\  j  e. 
dom  s )  -> 
( s `  j
)  e.  C )
8077, 79eqeltrd 2555 . . . . . . . . . . . 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 ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
8167, 80syldan 470 . . . . . . . . . . 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 )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
82 ablfac2.s . . . . . . . . . . . 12  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
83 fveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
w `  k )  =  ( w `  j ) )
8483oveq2d 6300 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  (
n  .x.  ( w `  k ) )  =  ( n  .x.  (
w `  j )
) )
8584mpteq2dv 4534 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
8685rneqd 5230 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
8786cbvmptv 4538 . . . . . . . . . . . 12  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8882, 87eqtri 2496 . . . . . . . . . . 11  |-  S  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8981, 88fmptd 6045 . . . . . . . . . 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 : dom  w --> C )
90 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  G dom DProd  s )
9190adantr 465 . . . . . . . . . . 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 ) ) )  ->  G dom DProd  s )
9265raleqdv 3064 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 ) ) )
9368, 92mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 ) )
94 mpteq12 4526 . . . . . . . . . . . . . 14  |-  ( ( 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
) ) )
9565, 93, 94syl2anc 661 . . . . . . . . . . . . 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 ) ) )  ->  ( k  e.  dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9682, 95syl5eq 2520 . . . . . . . . . . . 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  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
97 dprdf 16842 . . . . . . . . . . . . . 14  |-  ( G dom DProd  s  ->  s : dom  s --> (SubGrp `  G ) )
9891, 97syl 16 . . . . . . . . . . . . 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 ) ) )  ->  s : dom  s --> (SubGrp `  G )
)
9998feqmptd 5920 . . . . . . . . . . . 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  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
10096, 99eqtr4d 2511 . . . . . . . . . . 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  =  s )
10191, 100breqtrrd 4473 . . . . . . . . . 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 ) ) )  ->  G dom DProd  S )
102100oveq2d 6300 . . . . . . . . . . 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 ) ) )  ->  ( G DProd  S )  =  ( G DProd 
s ) )
103 simplrr 760 . . . . . . . . . . 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 ) ) )  ->  ( G DProd  s )  =  B )
104102, 103eqtrd 2508 . . . . . . . . . 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 ) ) )  ->  ( G DProd  S )  =  B )
10589, 101, 1043jca 1176 . . . . . . . . 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 : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
10662, 105jca 532 . . . . . . . 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  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) )
107106ex 434 . . . . . . 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  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) ) )
108107eximdv 1686 . . . . . 6  |-  ( ( ( 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 ) ) ) )
10956, 108mpd 15 . . . . 5  |-  ( ( ( 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 ) ) )
110 df-rex 2820 . . . . 5  |-  ( 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 ) ) )
111109, 110sylibr 212 . . . 4  |-  ( ( ( 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 ) )
112111ex 434 . . 3  |-  ( (
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 ) ) )
113112rexlimdva 2955 . 2  |-  ( ph  ->  ( E. 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 ) ) )
1145, 113mpd 15 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 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   0cc0 9492   ZZcz 10864  ..^cfzo 11792   #chash 12373  Word cword 12500   Basecbs 14490   ↾s cress 14491   Grpcgrp 15727  .gcmg 15731  SubGrpcsubg 16000   pGrp cpgp 16357   Abelcabl 16605  CycGrpccyg 16683   DProd cdprd 16827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-rpss 6564  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-eqg 16005  df-ghm 16070  df-gim 16112  df-ga 16133  df-cntz 16160  df-oppg 16186  df-od 16359  df-gex 16360  df-pgp 16361  df-lsm 16462  df-pj1 16463  df-cmn 16606  df-abl 16607  df-cyg 16684  df-dprd 16829
This theorem is referenced by:  dchrpt  23298
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