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Theorem pgpfaclem1 15316
Description: Lemma for pgpfac 15319. (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 )
pgpfac.2  |-  ( ph  ->  S  e. Word  C )
pgpfac.4  |-  ( ph  ->  G dom DProd  S )
pgpfac.5  |-  ( ph  ->  ( G DProd  S )  =  W )
pgpfac.t  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
Assertion
Ref Expression
pgpfaclem1  |-  ( 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    T, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    S( t, s, r)    T( t, 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 pgpfaclem1
StepHypRef Expression
1 pgpfac.t . . 3  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
2 pgpfac.2 . . 3  |-  ( ph  ->  S  e. Word  C )
3 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 pgpfac.h . . . . . . . . . 10  |-  H  =  ( Gs  U )
54subggrp 14624 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  H  e.  Grp )
63, 5syl 15 . . . . . . . 8  |-  ( ph  ->  H  e.  Grp )
7 eqid 2283 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
87subgacs 14652 . . . . . . . 8  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
9 acsmre 13554 . . . . . . . 8  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
106, 8, 93syl 18 . . . . . . 7  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
11 pgpfac.x . . . . . . . 8  |-  ( ph  ->  X  e.  U )
124subgbas 14625 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
133, 12syl 15 . . . . . . . 8  |-  ( ph  ->  U  =  ( Base `  H ) )
1411, 13eleqtrd 2359 . . . . . . 7  |-  ( ph  ->  X  e.  ( Base `  H ) )
15 pgpfac.k . . . . . . . 8  |-  K  =  (mrCls `  (SubGrp `  H
) )
1615mrcsncl 13514 . . . . . . 7  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
1710, 14, 16syl2anc 642 . . . . . 6  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
184subsubg 14640 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
193, 18syl 15 . . . . . 6  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
2017, 19mpbid 201 . . . . 5  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) )
2120simpld 445 . . . 4  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  G ) )
224oveq1i 5868 . . . . . . 7  |-  ( Hs  ( K `  { X } ) )  =  ( ( Gs  U )s  ( K `  { X } ) )
2320simprd 449 . . . . . . . 8  |-  ( ph  ->  ( K `  { X } )  C_  U
)
24 ressabs 13206 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
)  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
253, 23, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `  { X } ) ) )
2622, 25syl5eq 2327 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
277, 15cycsubgcyg2 15188 . . . . . . 7  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( Hs  ( K `  { X } ) )  e. CycGrp )
286, 14, 27syl2anc 642 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  e. CycGrp )
2926, 28eqeltrrd 2358 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e. CycGrp )
30 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
31 pgpprm 14904 . . . . . . 7  |-  ( P pGrp 
G  ->  P  e.  Prime )
3230, 31syl 15 . . . . . 6  |-  ( ph  ->  P  e.  Prime )
33 subgpgp 14908 . . . . . . 7  |-  ( ( P pGrp  G  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  ( K `  { X } ) ) )
3430, 21, 33syl2anc 642 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  ( K `
 { X }
) ) )
35 brelrng 4908 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( Gs  ( K `  { X } ) )  e. CycGrp  /\  P pGrp  ( Gs  ( K `  { X } ) ) )  ->  ( Gs  ( K `
 { X }
) )  e.  ran pGrp  )
3632, 29, 34, 35syl3anc 1182 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  ran pGrp  )
37 elin 3358 . . . . 5  |-  ( ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( ( Gs  ( K `  { X } ) )  e. CycGrp  /\  ( Gs  ( K `  { X } ) )  e.  ran pGrp  ) )
3829, 36, 37sylanbrc 645 . . . 4  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) )
39 oveq2 5866 . . . . . 6  |-  ( r  =  ( K `  { X } )  -> 
( Gs  r )  =  ( Gs  ( K `  { X } ) ) )
4039eleq1d 2349 . . . . 5  |-  ( r  =  ( K `  { X } )  -> 
( ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <-> 
( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
41 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
4240, 41elrab2 2925 . . . 4  |-  ( ( K `  { X } )  e.  C  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
4321, 38, 42sylanbrc 645 . . 3  |-  ( ph  ->  ( K `  { X } )  e.  C
)
441, 2, 43cats1cld 11505 . 2  |-  ( ph  ->  T  e. Word  C )
45 wrdf 11419 . . . . 5  |-  ( T  e. Word  C  ->  T : ( 0..^ (
# `  T )
) --> C )
4644, 45syl 15 . . . 4  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> C )
47 ssrab2 3258 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  G )
4841, 47eqsstri 3208 . . . 4  |-  C  C_  (SubGrp `  G )
49 fss 5397 . . . 4  |-  ( ( T : ( 0..^ ( # `  T
) ) --> C  /\  C  C_  (SubGrp `  G
) )  ->  T : ( 0..^ (
# `  T )
) --> (SubGrp `  G )
)
5046, 48, 49sylancl 643 . . 3  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> (SubGrp `  G ) )
51 fzodisj 10900 . . . 4  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  1 ) ) )  =  (/)
52 lencl 11421 . . . . . . . 8  |-  ( S  e. Word  C  ->  ( # `
 S )  e. 
NN0 )
532, 52syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  S
)  e.  NN0 )
5453nn0zd 10115 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ZZ )
55 fzosn 10912 . . . . . 6  |-  ( (
# `  S )  e.  ZZ  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) )  =  { (
# `  S ) } )
5654, 55syl 15 . . . . 5  |-  ( ph  ->  ( ( # `  S
)..^ ( ( # `  S )  +  1 ) )  =  {
( # `  S ) } )
5756ineq2d 3370 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) ) )  =  ( ( 0..^ ( # `  S ) )  i^i 
{ ( # `  S
) } ) )
5851, 57syl5reqr 2330 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  { ( # `
 S ) } )  =  (/) )
591fveq2i 5528 . . . . . . 7  |-  ( # `  T )  =  (
# `  ( S concat  <" ( K `  { X } ) "> ) )
6043s1cld 11442 . . . . . . . 8  |-  ( ph  ->  <" ( K `
 { X }
) ">  e. Word  C )
61 ccatlen 11430 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C )  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
622, 60, 61syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
6359, 62syl5eq 2327 . . . . . 6  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
64 s1len 11444 . . . . . . 7  |-  ( # `  <" ( K `
 { X }
) "> )  =  1
6564oveq2i 5869 . . . . . 6  |-  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) )  =  ( ( # `  S
)  +  1 )
6663, 65syl6eq 2331 . . . . 5  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  1 ) )
6766oveq2d 5874 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( 0..^ ( (
# `  S )  +  1 ) ) )
68 nn0uz 10262 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
6953, 68syl6eleq 2373 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
70 fzosplitsn 10920 . . . . 5  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7169, 70syl 15 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7267, 71eqtrd 2315 . . 3  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } ) )
73 eqid 2283 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
74 eqid 2283 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
75 pgpfac.4 . . . 4  |-  ( ph  ->  G dom DProd  S )
76 cats1un 11476 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  ( K `  { X } )  e.  C
)  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
772, 43, 76syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } ) )
781, 77syl5eq 2327 . . . . . 6  |-  ( ph  ->  T  =  ( S  u.  { <. ( # `
 S ) ,  ( K `  { X } ) >. } ) )
7978reseq1d 4954 . . . . 5  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) ) )
80 wrdf 11419 . . . . . . 7  |-  ( S  e. Word  C  ->  S : ( 0..^ (
# `  S )
) --> C )
81 ffn 5389 . . . . . . 7  |-  ( S : ( 0..^ (
# `  S )
) --> C  ->  S  Fn  ( 0..^ ( # `  S ) ) )
822, 80, 813syl 18 . . . . . 6  |-  ( ph  ->  S  Fn  ( 0..^ ( # `  S
) ) )
83 fzonel 10887 . . . . . 6  |-  -.  ( # `
 S )  e.  ( 0..^ ( # `  S ) )
84 fsnunres 5721 . . . . . 6  |-  ( ( S  Fn  ( 0..^ ( # `  S
) )  /\  -.  ( # `  S )  e.  ( 0..^ (
# `  S )
) )  ->  (
( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8582, 83, 84sylancl 643 . . . . 5  |-  ( ph  ->  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8679, 85eqtrd 2315 . . . 4  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  S )
8775, 86breqtrrd 4049 . . 3  |-  ( ph  ->  G dom DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )
88 fvex 5539 . . . . . 6  |-  ( # `  S )  e.  _V
89 dprdsn 15271 . . . . . 6  |-  ( ( ( # `  S
)  e.  _V  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9088, 21, 89sylancr 644 . . . . 5  |-  ( ph  ->  ( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9190simpld 445 . . . 4  |-  ( ph  ->  G dom DProd  { <. ( # `
 S ) ,  ( K `  { X } ) >. } )
92 ffn 5389 . . . . . . 7  |-  ( T : ( 0..^ (
# `  T )
) --> C  ->  T  Fn  ( 0..^ ( # `  T ) ) )
9344, 45, 923syl 18 . . . . . 6  |-  ( ph  ->  T  Fn  ( 0..^ ( # `  T
) ) )
94 ssun2 3339 . . . . . . . 8  |-  { (
# `  S ) }  C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } )
9588snss 3748 . . . . . . . 8  |-  ( (
# `  S )  e.  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } )  <->  { ( # `
 S ) } 
C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
9694, 95mpbir 200 . . . . . . 7  |-  ( # `  S )  e.  ( ( 0..^ ( # `  S ) )  u. 
{ ( # `  S
) } )
9796, 72syl5eleqr 2370 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ( 0..^ ( # `  T
) ) )
98 fnressn 5705 . . . . . 6  |-  ( ( T  Fn  ( 0..^ ( # `  T
) )  /\  ( # `
 S )  e.  ( 0..^ ( # `  T ) ) )  ->  ( T  |`  { ( # `  S
) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
9993, 97, 98syl2anc 642 . . . . 5  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
1001fveq1i 5526 . . . . . . . . 9  |-  ( T `
 ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )
10153nn0cnd 10020 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  S
)  e.  CC )
102101addid2d 9013 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  +  (
# `  S )
)  =  ( # `  S ) )
103102eqcomd 2288 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  =  ( 0  +  ( # `  S
) ) )
104103fveq2d 5529 . . . . . . . . 9  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) ) )
105100, 104syl5eq 2327 . . . . . . . 8  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  (
0  +  ( # `  S ) ) ) )
106 1nn 9757 . . . . . . . . . . . 12  |-  1  e.  NN
107106a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  NN )
10864, 107syl5eqel 2367 . . . . . . . . . 10  |-  ( ph  ->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
109 lbfzo0 10903 . . . . . . . . . 10  |-  ( 0  e.  ( 0..^ (
# `  <" ( K `  { X } ) "> ) )  <->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
110108, 109sylibr 203 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )
111 ccatval3 11433 . . . . . . . . 9  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C  /\  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )  -> 
( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
1122, 60, 110, 111syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
113 fvex 5539 . . . . . . . . 9  |-  ( K `
 { X }
)  e.  _V
114 s1fv 11446 . . . . . . . . 9  |-  ( ( K `  { X } )  e.  _V  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
115113, 114mp1i 11 . . . . . . . 8  |-  ( ph  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
116105, 112, 1153eqtrd 2319 . . . . . . 7  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( K `  { X } ) )
117116opeq2d 3803 . . . . . 6  |-  ( ph  -> 
<. ( # `  S
) ,  ( T `
 ( # `  S
) ) >.  =  <. (
# `  S ) ,  ( K `  { X } ) >.
)
118117sneqd 3653 . . . . 5  |-  ( ph  ->  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. }  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
11999, 118eqtrd 2315 . . . 4  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
12091, 119breqtrrd 4049 . . 3  |-  ( ph  ->  G dom DProd  ( T  |` 
{ ( # `  S
) } ) )
121 pgpfac.g . . . 4  |-  ( ph  ->  G  e.  Abel )
122 dprdsubg 15259 . . . . 5  |-  ( G dom DProd  ( T  |`  ( 0..^ ( # `  S
) ) )  -> 
( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
12387, 122syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
124 dprdsubg 15259 . . . . 5  |-  ( G dom DProd  ( T  |`  { ( # `  S
) } )  -> 
( G DProd  ( T  |` 
{ ( # `  S
) } ) )  e.  (SubGrp `  G
) )
125120, 124syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  e.  (SubGrp `  G
) )
12673, 121, 123, 125ablcntzd 15149 . . 3  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( T  |`  { (
# `  S ) } ) ) ) )
127 pgpfac.i . . . 4  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
12886oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  ( G DProd 
S ) )
129 pgpfac.5 . . . . . . 7  |-  ( ph  ->  ( G DProd  S )  =  W )
130128, 129eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  W )
131119oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
13290simprd 449 . . . . . . 7  |-  ( ph  ->  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) )
133131, 132eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( K `  { X } ) )
134130, 133ineq12d 3371 . . . . 5  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W  i^i  ( K `  { X } ) ) )
135 incom 3361 . . . . 5  |-  ( W  i^i  ( K `  { X } ) )  =  ( ( K `
 { X }
)  i^i  W )
136134, 135syl6eq 2331 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( ( K `  { X } )  i^i  W
) )
1374, 74subg0 14627 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1383, 137syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
139 pgpfac.0 . . . . . 6  |-  .0.  =  ( 0g `  H )
140138, 139syl6eqr 2333 . . . . 5  |-  ( ph  ->  ( 0g `  G
)  =  .0.  )
141140sneqd 3653 . . . 4  |-  ( ph  ->  { ( 0g `  G ) }  =  {  .0.  } )
142127, 136, 1413eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  { ( 0g `  G ) } )
14350, 58, 72, 73, 74, 87, 120, 126, 142dmdprdsplit2 15281 . 2  |-  ( ph  ->  G dom DProd  T )
144 eqid 2283 . . . . 5  |-  ( LSSum `  G )  =  (
LSSum `  G )
14550, 58, 72, 144, 143dprdsplit 15283 . . . 4  |-  ( ph  ->  ( G DProd  T )  =  ( ( G DProd 
( T  |`  (
0..^ ( # `  S
) ) ) ) ( LSSum `  G )
( G DProd  ( T  |` 
{ ( # `  S
) } ) ) ) )
146130, 133oveq12d 5876 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) ) (
LSSum `  G ) ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W ( LSSum `  G )
( K `  { X } ) ) )
147130, 123eqeltrrd 2358 . . . . 5  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
148144lsmcom 15150 . . . . 5  |-  ( ( G  e.  Abel  /\  W  e.  (SubGrp `  G )  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
149121, 147, 21, 148syl3anc 1182 . . . 4  |-  ( ph  ->  ( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
150145, 146, 1493eqtrd 2319 . . 3  |-  ( ph  ->  ( G DProd  T )  =  ( ( K `
 { X }
) ( LSSum `  G
) W ) )
151 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
1527subgss 14622 . . . . . 6  |-  ( W  e.  (SubGrp `  H
)  ->  W  C_  ( Base `  H ) )
153151, 152syl 15 . . . . 5  |-  ( ph  ->  W  C_  ( Base `  H ) )
154153, 13sseqtr4d 3215 . . . 4  |-  ( ph  ->  W  C_  U )
155 pgpfac.l . . . . 5  |-  .(+)  =  (
LSSum `  H )
1564, 144, 155subglsm 14982 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U  /\  W  C_  U )  ->  ( ( K `
 { X }
) ( LSSum `  G
) W )  =  ( ( K `  { X } )  .(+)  W ) )
1573, 23, 154, 156syl3anc 1182 . . 3  |-  ( ph  ->  ( ( K `  { X } ) (
LSSum `  G ) W )  =  ( ( K `  { X } )  .(+)  W ) )
158 pgpfac.s . . 3  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
159150, 157, 1583eqtrd 2319 . 2  |-  ( ph  ->  ( G DProd  T )  =  U )
160 breq2 4027 . . . 4  |-  ( s  =  T  ->  ( G dom DProd  s  <->  G dom DProd  T ) )
161 oveq2 5866 . . . . 5  |-  ( s  =  T  ->  ( G DProd  s )  =  ( G DProd  T ) )
162161eqeq1d 2291 . . . 4  |-  ( s  =  T  ->  (
( G DProd  s )  =  U  <->  ( G DProd  T
)  =  U ) )
163160, 162anbi12d 691 . . 3  |-  ( s  =  T  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  T  /\  ( G DProd 
T )  =  U ) ) )
164163rspcev 2884 . 2  |-  ( ( T  e. Word  C  /\  ( G dom DProd  T  /\  ( G DProd  T )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
16544, 143, 159, 164syl12anc 1180 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230  ..^cfzo 10870   #chash 11337  Word cword 11403   concat cconcat 11404   <"cs1 11405   Primecprime 12758   Basecbs 13148   ↾s cress 13149   0gc0g 13400  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   odcod 14840  gExcgex 14841   pGrp cpgp 14842   LSSumclsm 14945   Abelcabel 15090  CycGrpccyg 15164   DProd cdprd 15231
This theorem is referenced by:  pgpfaclem2  15317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-od 14844  df-pgp 14846  df-lsm 14947  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233
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