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Theorem ablfaclem1 15336
Description: Lemma for ablfac 15339. (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
ablfaclem1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
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    U, g, s    g, G, p, r, s, w, x
Allowed substitution hints:    ph( g, r)    A( w, g, r)    C( r)    S( x, w, p)    U( x, w, r, p)    O( w, g, s, r)    W( g, s, r)

Proof of Theorem ablfaclem1
StepHypRef Expression
1 eqeq2 2305 . . . 4  |-  ( g  =  U  ->  (
( G DProd  s )  =  g  <->  ( G DProd  s
)  =  U ) )
21anbi2d 684 . . 3  |-  ( g  =  U  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) ) )
32rabbidv 2793 . 2  |-  ( g  =  U  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) } )
4 ablfac.w . 2  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
5 ablfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
6 fvex 5555 . . . . . 6  |-  (SubGrp `  G )  e.  _V
76rabex 4181 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  e.  _V
85, 7eqeltri 2366 . . . 4  |-  C  e. 
_V
9 wrdexg 11441 . . . 4  |-  ( C  e.  _V  -> Word  C  e. 
_V )
108, 9ax-mp 8 . . 3  |- Word  C  e. 
_V
1110rabex 4181 . 2  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  U ) }  e.  _V
123, 4, 11fvmpt 5618 1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    i^i cin 3164   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874   Fincfn 6879   ^cexp 11120   #chash 11353  Word cword 11419    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   ↾s cress 13165  SubGrpcsubg 14631   odcod 14856   pGrp cpgp 14858   Abelcabel 15106  CycGrpccyg 15180   DProd cdprd 15247
This theorem is referenced by:  ablfaclem2  15337  ablfaclem3  15338  ablfac  15339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-pm 6791  df-neg 9056  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-word 11425
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