MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfaclem3 Structured version   Unicode version

Theorem pgpfaclem3 17008
Description: Lemma for pgpfac 17009. (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 ) ) )
Assertion
Ref Expression
pgpfaclem3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    ph, t    B, s, t    U, r, s, t
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)

Proof of Theorem pgpfaclem3
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd0 12544 . . 3  |-  (/)  e. Word  C
2 pgpfac.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
3 ablgrp 16677 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 eqid 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
54dprd0 16952 . . . . . 6  |-  ( G  e.  Grp  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
62, 3, 53syl 20 . . . . 5  |-  ( ph  ->  ( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  {
( 0g `  G
) } ) )
76adantr 465 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
8 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
94subg0cl 16083 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  U
)
108, 9syl 16 . . . . . . . 8  |-  ( ph  ->  ( 0g `  G
)  e.  U )
1110adantr 465 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( 0g `  G )  e.  U )
12 eqid 2443 . . . . . . . . . . 11  |-  ( Gs  U )  =  ( Gs  U )
1312subgbas 16079 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  ( Gs  U
) ) )
148, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  ( Gs  U ) ) )
1514adantr 465 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  ( Base `  ( Gs  U ) ) )
1612subggrp 16078 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  ( Gs  U
)  e.  Grp )
178, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Gs  U )  e.  Grp )
18 grpmnd 15936 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Grp  ->  ( Gs  U )  e.  Mnd )
19 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  ( Gs  U ) )  =  ( Base `  ( Gs  U ) )
20 eqid 2443 . . . . . . . . . . 11  |-  (gEx `  ( Gs  U ) )  =  (gEx `  ( Gs  U
) )
2119, 20gex1 16485 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Mnd  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2217, 18, 213syl 20 . . . . . . . . 9  |-  ( ph  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2322biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( Base `  ( Gs  U ) )  ~~  1o )
2415, 23eqbrtrd 4457 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  ~~  1o )
25 en1eqsn 7751 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  U  /\  U  ~~  1o )  ->  U  =  { ( 0g `  G ) } )
2611, 24, 25syl2anc 661 . . . . . 6  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  { ( 0g `  G ) } )
2726eqeq2d 2457 . . . . 5  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G DProd  (/) )  =  U  <->  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
2827anbi2d 703 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) ) )
297, 28mpbird 232 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )
30 breq2 4441 . . . . 5  |-  ( s  =  (/)  ->  ( G dom DProd  s  <->  G dom DProd  (/) ) )
31 oveq2 6289 . . . . . 6  |-  ( s  =  (/)  ->  ( G DProd 
s )  =  ( G DProd  (/) ) )
3231eqeq1d 2445 . . . . 5  |-  ( s  =  (/)  ->  ( ( G DProd  s )  =  U  <->  ( G DProd  (/) )  =  U ) )
3330, 32anbi12d 710 . . . 4  |-  ( s  =  (/)  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) ) )
3433rspcev 3196 . . 3  |-  ( (
(/)  e. Word  C  /\  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
351, 29, 34sylancr 663 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
3612subgabl 16718 . . . . . 6  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  ( Gs  U
)  e.  Abel )
372, 8, 36syl2anc 661 . . . . 5  |-  ( ph  ->  ( Gs  U )  e.  Abel )
38 pgpfac.f . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
39 pgpfac.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
4039subgss 16076 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
418, 40syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  B )
42 ssfi 7742 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
4338, 41, 42syl2anc 661 . . . . . . 7  |-  ( ph  ->  U  e.  Fin )
4414, 43eqeltrrd 2532 . . . . . 6  |-  ( ph  ->  ( Base `  ( Gs  U ) )  e. 
Fin )
4519, 20gexcl2 16483 . . . . . 6  |-  ( ( ( Gs  U )  e.  Grp  /\  ( Base `  ( Gs  U ) )  e. 
Fin )  ->  (gEx `  ( Gs  U ) )  e.  NN )
4617, 44, 45syl2anc 661 . . . . 5  |-  ( ph  ->  (gEx `  ( Gs  U
) )  e.  NN )
47 eqid 2443 . . . . . 6  |-  ( od
`  ( Gs  U ) )  =  ( od
`  ( Gs  U ) )
4819, 20, 47gexex 16733 . . . . 5  |-  ( ( ( Gs  U )  e.  Abel  /\  (gEx `  ( Gs  U
) )  e.  NN )  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
4937, 46, 48syl2anc 661 . . . 4  |-  ( ph  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
5049adantr 465 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. x  e.  ( Base `  ( Gs  U ) ) ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
51 eqid 2443 . . . . 5  |-  (mrCls `  (SubGrp `  ( Gs  U ) ) )  =  (mrCls `  (SubGrp `  ( Gs  U
) ) )
52 eqid 2443 . . . . 5  |-  ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  =  ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )
53 eqid 2443 . . . . 5  |-  ( 0g
`  ( Gs  U ) )  =  ( 0g
`  ( Gs  U ) )
54 eqid 2443 . . . . 5  |-  ( LSSum `  ( Gs  U ) )  =  ( LSSum `  ( Gs  U
) )
55 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
56 subgpgp 16491 . . . . . . 7  |-  ( ( P pGrp  G  /\  U  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  U ) )
5755, 8, 56syl2anc 661 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  U ) )
5857ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  P pGrp  ( Gs  U ) )
5937ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Gs  U )  e.  Abel )
6044ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Base `  ( Gs  U
) )  e.  Fin )
61 simprr 757 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
62 simprl 756 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
6351, 52, 19, 47, 20, 53, 54, 58, 59, 60, 61, 62pgpfac1 17005 . . . 4  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. w  e.  (SubGrp `  ( Gs  U ) ) ( ( ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )  i^i  w )  =  {
( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) )
64 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
652ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  G  e.  Abel )
6655ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  P pGrp  G )
6738ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  B  e.  Fin )
688ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  e.  (SubGrp `  G )
)
69 pgpfac.a . . . . . 6  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
7069ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  A. t  e.  (SubGrp `  G )
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) ) )
71 simpllr 760 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (gEx `  ( Gs  U ) )  =/=  1 )
72 simplrl 761 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
7368, 13syl 16 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  =  ( Base `  ( Gs  U ) ) )
7472, 73eleqtrrd 2534 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  U )
75 simplrr 762 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
76 simprl 756 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  w  e.  (SubGrp `  ( Gs  U
) ) )
77 simprrl 765 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
)  i^i  w )  =  { ( 0g `  ( Gs  U ) ) } )
78 simprrr 766 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  ( Base `  ( Gs  U ) ) )
7978, 73eqtr4d 2487 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  U )
8039, 64, 65, 66, 67, 68, 70, 12, 51, 47, 20, 53, 54, 71, 74, 75, 76, 77, 79pgpfaclem2 17007 . . . 4  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8163, 80rexlimddv 2939 . . 3  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
8250, 81rexlimddv 2939 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8335, 82pm2.61dane 2761 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3770   {csn 4014   class class class wbr 4437   dom cdm 4989   ran crn 4990   ` cfv 5578  (class class class)co 6281   1oc1o 7125    ~~ cen 7515   Fincfn 7518   1c1 9496   NNcn 10542  Word cword 12513   Basecbs 14509   ↾s cress 14510   0gc0g 14714  mrClscmrc 14857   Mndcmnd 15793   Grpcgrp 15927  SubGrpcsubg 16069   odcod 16423  gExcgex 16424   pGrp cpgp 16425   LSSumclsm 16528   Abelcabl 16673  CycGrpccyg 16754   DProd cdprd 16898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-rpss 6565  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-dvds 13864  df-gcd 14022  df-prm 14095  df-pc 14238  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-gsum 14717  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-eqg 16074  df-ghm 16139  df-gim 16181  df-ga 16202  df-cntz 16229  df-oppg 16255  df-od 16427  df-gex 16428  df-pgp 16429  df-lsm 16530  df-pj1 16531  df-cmn 16674  df-abl 16675  df-cyg 16755  df-dprd 16900
This theorem is referenced by:  pgpfac  17009
  Copyright terms: Public domain W3C validator