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Theorem ablfaclem1 16938
Description: Lemma for ablfac 16941. (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 2482 . . . 4  |-  ( g  =  U  ->  (
( G DProd  s )  =  g  <->  ( G DProd  s
)  =  U ) )
21anbi2d 703 . . 3  |-  ( g  =  U  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) ) )
32rabbidv 3105 . 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 5876 . . . . . 6  |-  (SubGrp `  G )  e.  _V
76rabex 4598 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  e.  _V
85, 7eqeltri 2551 . . . 4  |-  C  e. 
_V
9 wrdexg 12523 . . . 4  |-  ( C  e.  _V  -> Word  C  e. 
_V )
108, 9ax-mp 5 . . 3  |- Word  C  e. 
_V
1110rabex 4598 . 2  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  U ) }  e.  _V
123, 4, 11fvmpt 5950 1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000   ` cfv 5588  (class class class)co 6284   Fincfn 7516   ^cexp 12134   #chash 12373  Word cword 12500    || cdivides 13847   Primecprime 14076    pCnt cpc 14219   Basecbs 14490   ↾s cress 14491  SubGrpcsubg 16000   odcod 16355   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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-ral 2819  df-rex 2820  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-pm 7423  df-neg 9808  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-word 12508
This theorem is referenced by:  ablfaclem2  16939  ablfaclem3  16940  ablfac  16941
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