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Theorem pgpfaclem2 17328
Description: Lemma for pgpfac 17330. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
pgpfac.b  |-  B  =  ( Base `  G
)
pgpfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
pgpfac.g  |-  ( ph  ->  G  e.  Abel )
pgpfac.p  |-  ( ph  ->  P pGrp  G )
pgpfac.f  |-  ( ph  ->  B  e.  Fin )
pgpfac.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac.a  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
pgpfac.h  |-  H  =  ( Gs  U )
pgpfac.k  |-  K  =  (mrCls `  (SubGrp `  H
) )
pgpfac.o  |-  O  =  ( od `  H
)
pgpfac.e  |-  E  =  (gEx `  H )
pgpfac.0  |-  .0.  =  ( 0g `  H )
pgpfac.l  |-  .(+)  =  (
LSSum `  H )
pgpfac.1  |-  ( ph  ->  E  =/=  1 )
pgpfac.x  |-  ( ph  ->  X  e.  U )
pgpfac.oe  |-  ( ph  ->  ( O `  X
)  =  E )
pgpfac.w  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
pgpfac.i  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
pgpfac.s  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
Assertion
Ref Expression
pgpfaclem2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    K, r, s    ph, t    B, s, t    U, r, s, t    W, s, t    X, r, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    E( t, s, r)    H( t, s, r)    K( t)    O( t, s, r)    W( r)    X( t)    .0. ( t, s, r)

Proof of Theorem pgpfaclem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
2 pgpfac.u . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 pgpfac.h . . . . . . . 8  |-  H  =  ( Gs  U )
43subsubg 16423 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
52, 4syl 16 . . . . . 6  |-  ( ph  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
61, 5mpbid 210 . . . . 5  |-  ( ph  ->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) )
76simpld 457 . . . 4  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
8 pgpfac.a . . . 4  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
96simprd 461 . . . . 5  |-  ( ph  ->  W  C_  U )
10 pgpfac.f . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
11 pgpfac.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
1211subgss 16401 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
132, 12syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
14 ssfi 7733 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
1510, 13, 14syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  U  e.  Fin )
16 ssfi 7733 . . . . . . . . . 10  |-  ( ( U  e.  Fin  /\  W  C_  U )  ->  W  e.  Fin )
1715, 9, 16syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  W  e.  Fin )
18 hashcl 12410 . . . . . . . . 9  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
1917, 18syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
2019nn0red 10849 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  e.  RR )
21 pgpfac.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  H )
22 fvex 5858 . . . . . . . . . . . 12  |-  ( 0g
`  H )  e. 
_V
2321, 22eqeltri 2538 . . . . . . . . . . 11  |-  .0.  e.  _V
24 hashsng 12421 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( # `  {  .0.  } )  =  1 )
2523, 24ax-mp 5 . . . . . . . . . 10  |-  ( # `  {  .0.  } )  =  1
26 subgrcl 16405 . . . . . . . . . . . . . . . 16  |-  ( W  e.  (SubGrp `  H
)  ->  H  e.  Grp )
27 eqid 2454 . . . . . . . . . . . . . . . . 17  |-  ( Base `  H )  =  (
Base `  H )
2827subgacs 16435 . . . . . . . . . . . . . . . 16  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
29 acsmre 15141 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
301, 26, 28, 294syl 21 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
31 pgpfac.k . . . . . . . . . . . . . . 15  |-  K  =  (mrCls `  (SubGrp `  H
) )
3230, 31mrcssvd 15112 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  { X } )  C_  ( Base `  H ) )
333subgbas 16404 . . . . . . . . . . . . . . 15  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
342, 33syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  =  ( Base `  H ) )
3532, 34sseqtr4d 3526 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  { X } )  C_  U
)
36 ssfi 7733 . . . . . . . . . . . . 13  |-  ( ( U  e.  Fin  /\  ( K `  { X } )  C_  U
)  ->  ( K `  { X } )  e.  Fin )
3715, 35, 36syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( K `  { X } )  e.  Fin )
38 pgpfac.x . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  U )
3938, 34eleqtrd 2544 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( Base `  H ) )
4031mrcsncl 15101 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
4130, 39, 40syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
4221subg0cl 16408 . . . . . . . . . . . . . . 15  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  .0.  e.  ( K `  { X } ) )
4341, 42syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  .0.  e.  ( K `
 { X }
) )
4443snssd 4161 . . . . . . . . . . . . 13  |-  ( ph  ->  {  .0.  }  C_  ( K `  { X } ) )
4539snssd 4161 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { X }  C_  ( Base `  H )
)
4630, 31, 45mrcssidd 15114 . . . . . . . . . . . . . 14  |-  ( ph  ->  { X }  C_  ( K `  { X } ) )
47 snssg 4149 . . . . . . . . . . . . . . 15  |-  ( X  e.  U  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4838, 47syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4946, 48mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  ( K `
 { X }
) )
50 pgpfac.oe . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( O `  X
)  =  E )
51 pgpfac.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  =/=  1 )
5250, 51eqnetrd 2747 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  X
)  =/=  1 )
53 pgpfac.o . . . . . . . . . . . . . . . . . 18  |-  O  =  ( od `  H
)
5453, 21od1 16780 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  ( O `  .0.  )  =  1 )
551, 26, 543syl 20 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( O `  .0.  )  =  1 )
56 elsni 4041 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  {  .0.  }  ->  X  =  .0.  )
5756fveq2d 5852 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  {  .0.  }  ->  ( O `  X
)  =  ( O `
 .0.  ) )
5857eqeq1d 2456 . . . . . . . . . . . . . . . 16  |-  ( X  e.  {  .0.  }  ->  ( ( O `  X )  =  1  <-> 
( O `  .0.  )  =  1 ) )
5955, 58syl5ibrcom 222 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  {  .0.  }  ->  ( O `  X )  =  1 ) )
6059necon3ad 2664 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( O `  X )  =/=  1  ->  -.  X  e.  {  .0.  } ) )
6152, 60mpd 15 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  {  .0.  } )
6244, 49, 61ssnelpssd 3879 . . . . . . . . . . . 12  |-  ( ph  ->  {  .0.  }  C.  ( K `  { X } ) )
63 php3 7696 . . . . . . . . . . . 12  |-  ( ( ( K `  { X } )  e.  Fin  /\ 
{  .0.  }  C.  ( K `  { X } ) )  ->  {  .0.  }  ~<  ( K `  { X } ) )
6437, 62, 63syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  {  .0.  }  ~<  ( K `  { X } ) )
65 snfi 7589 . . . . . . . . . . . 12  |-  {  .0.  }  e.  Fin
66 hashsdom 12432 . . . . . . . . . . . 12  |-  ( ( {  .0.  }  e.  Fin  /\  ( K `  { X } )  e. 
Fin )  ->  (
( # `  {  .0.  } )  <  ( # `  ( K `  { X } ) )  <->  {  .0.  } 
~<  ( K `  { X } ) ) )
6765, 37, 66sylancr 661 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) )  <->  {  .0.  }  ~<  ( K `  { X } ) ) )
6864, 67mpbird 232 . . . . . . . . . 10  |-  ( ph  ->  ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) ) )
6925, 68syl5eqbrr 4473 . . . . . . . . 9  |-  ( ph  ->  1  <  ( # `  ( K `  { X } ) ) )
70 1red 9600 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
71 hashcl 12410 . . . . . . . . . . . 12  |-  ( ( K `  { X } )  e.  Fin  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7237, 71syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7372nn0red 10849 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e.  RR )
7421subg0cl 16408 . . . . . . . . . . . . 13  |-  ( W  e.  (SubGrp `  H
)  ->  .0.  e.  W )
75 ne0i 3789 . . . . . . . . . . . . 13  |-  (  .0. 
e.  W  ->  W  =/=  (/) )
761, 74, 753syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  W  =/=  (/) )
77 hashnncl 12419 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
7817, 77syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
7976, 78mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  NN )
8079nngt0d 10575 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( # `  W ) )
81 ltmul1 10388 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( # `  ( K `
 { X }
) )  e.  RR  /\  ( ( # `  W
)  e.  RR  /\  0  <  ( # `  W
) ) )  -> 
( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8270, 73, 20, 80, 81syl112anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8369, 82mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) )
8420recnd 9611 . . . . . . . . 9  |-  ( ph  ->  ( # `  W
)  e.  CC )
8584mulid2d 9603 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  =  ( # `  W
) )
86 pgpfac.l . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  H )
87 eqid 2454 . . . . . . . . . 10  |-  (Cntz `  H )  =  (Cntz `  H )
88 pgpfac.i . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
89 pgpfac.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  Abel )
903subgabl 17043 . . . . . . . . . . . 12  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
9189, 2, 90syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  Abel )
9287, 91, 41, 1ablcntzd 17062 . . . . . . . . . 10  |-  ( ph  ->  ( K `  { X } )  C_  (
(Cntz `  H ) `  W ) )
9386, 21, 87, 41, 1, 88, 92, 37, 17lsmhash 16922 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( (
# `  ( K `  { X } ) )  x.  ( # `  W ) ) )
94 pgpfac.s . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
9594fveq2d 5852 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( # `  U ) )
9693, 95eqtr3d 2497 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) )  =  (
# `  U )
)
9783, 85, 963brtr3d 4468 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  <  ( # `  U
) )
9820, 97ltned 9710 . . . . . 6  |-  ( ph  ->  ( # `  W
)  =/=  ( # `  U ) )
99 fveq2 5848 . . . . . . 7  |-  ( W  =  U  ->  ( # `
 W )  =  ( # `  U
) )
10099necon3i 2694 . . . . . 6  |-  ( (
# `  W )  =/=  ( # `  U
)  ->  W  =/=  U )
10198, 100syl 16 . . . . 5  |-  ( ph  ->  W  =/=  U )
102 df-pss 3477 . . . . 5  |-  ( W 
C.  U  <->  ( W  C_  U  /\  W  =/= 
U ) )
1039, 101, 102sylanbrc 662 . . . 4  |-  ( ph  ->  W  C.  U )
104 psseq1 3577 . . . . . 6  |-  ( t  =  W  ->  (
t  C.  U  <->  W  C.  U
) )
105 eqeq2 2469 . . . . . . . 8  |-  ( t  =  W  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  W ) )
106105anbi2d 701 . . . . . . 7  |-  ( t  =  W  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  W ) ) )
107106rexbidv 2965 . . . . . 6  |-  ( t  =  W  ->  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  W ) ) )
108104, 107imbi12d 318 . . . . 5  |-  ( t  =  W  ->  (
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  <-> 
( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
109108rspcv 3203 . . . 4  |-  ( W  e.  (SubGrp `  G
)  ->  ( A. t  e.  (SubGrp `  G
) ( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  ->  ( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
1107, 8, 103, 109syl3c 61 . . 3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) )
111 breq2 4443 . . . . 5  |-  ( s  =  a  ->  ( G dom DProd  s  <->  G dom DProd  a ) )
112 oveq2 6278 . . . . . 6  |-  ( s  =  a  ->  ( G DProd  s )  =  ( G DProd  a ) )
113112eqeq1d 2456 . . . . 5  |-  ( s  =  a  ->  (
( G DProd  s )  =  W  <->  ( G DProd  a
)  =  W ) )
114111, 113anbi12d 708 . . . 4  |-  ( s  =  a  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  ( G dom DProd  a  /\  ( G DProd 
a )  =  W ) ) )
115114cbvrexv 3082 . . 3  |-  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  E. a  e. Word  C ( G dom DProd  a  /\  ( G DProd  a
)  =  W ) )
116110, 115sylib 196 . 2  |-  ( ph  ->  E. a  e. Word  C
( G dom DProd  a  /\  ( G DProd  a )  =  W ) )
117 pgpfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
11889adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G  e.  Abel )
119 pgpfac.p . . . 4  |-  ( ph  ->  P pGrp  G )
120119adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  P pGrp  G )
12110adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  B  e.  Fin )
1222adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  U  e.  (SubGrp `  G
) )
1238adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
124 pgpfac.e . . 3  |-  E  =  (gEx `  H )
12551adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E  =/=  1 )
12638adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  X  e.  U )
12750adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( O `  X
)  =  E )
1281adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  W  e.  (SubGrp `  H
) )
12988adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
13094adantr 463 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  .(+)  W )  =  U )
131 simprl 754 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
a  e. Word  C )
132 simprrl 763 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G dom DProd  a )
133 simprrr 764 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( G DProd  a )  =  W )
134 eqid 2454 . . 3  |-  ( a ++ 
<" ( K `  { X } ) "> )  =  ( a ++  <" ( K `
 { X }
) "> )
13511, 117, 118, 120, 121, 122, 123, 3, 31, 53, 124, 21, 86, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134pgpfaclem1 17327 . 2  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
136116, 135rexlimddv 2950 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3783   {csn 4016   class class class wbr 4439   dom cdm 4988   ran crn 4989   ` cfv 5570  (class class class)co 6270    ~< csdm 7508   Fincfn 7509   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617   NNcn 10531   NN0cn0 10791   #chash 12387  Word cword 12518   ++ cconcat 12520   <"cs1 12521   Basecbs 14716   ↾s cress 14717   0gc0g 14929  Moorecmre 15071  mrClscmrc 15072  ACScacs 15074   Grpcgrp 16252  SubGrpcsubg 16394  Cntzccntz 16552   odcod 16748  gExcgex 16749   pGrp cpgp 16750   LSSumclsm 16853   Abelcabl 16998  CycGrpccyg 17079   DProd cdprd 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-ghm 16464  df-gim 16506  df-cntz 16554  df-oppg 16580  df-od 16752  df-pgp 16754  df-lsm 16855  df-pj1 16856  df-cmn 16999  df-abl 17000  df-cyg 17080  df-dprd 17221
This theorem is referenced by:  pgpfaclem3  17329
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