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Theorem subgdmdprd 16949
Description: A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypothesis
Ref Expression
subgdprd.1  |-  H  =  ( Gs  A )
Assertion
Ref Expression
subgdmdprd  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )

Proof of Theorem subgdmdprd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldmdprd 16897 . . . 4  |-  Rel  dom DProd
21brrelex2i 5027 . . 3  |-  ( H dom DProd  S  ->  S  e. 
_V )
32a1i 11 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  ->  S  e.  _V ) )
41brrelex2i 5027 . . . 4  |-  ( G dom DProd  S  ->  S  e. 
_V )
54adantr 465 . . 3  |-  ( ( G dom DProd  S  /\  ran  S  C_  ~P A
)  ->  S  e.  _V )
65a1i 11 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( G dom DProd  S  /\  ran  S  C_ 
~P A )  ->  S  e.  _V )
)
7 ffvelrn 6010 . . . . . . . . . . . . . . . 16  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  x  e.  dom  S )  ->  ( S `  x )  e.  (SubGrp `  H ) )
87ad2ant2lr 747 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  e.  (SubGrp `  H ) )
9 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( Base `  H )  =  (
Base `  H )
109subgss 16071 . . . . . . . . . . . . . . 15  |-  ( ( S `  x )  e.  (SubGrp `  H
)  ->  ( S `  x )  C_  ( Base `  H ) )
118, 10syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  ( Base `  H ) )
12 subgdprd.1 . . . . . . . . . . . . . . . 16  |-  H  =  ( Gs  A )
1312subgbas 16074 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1413ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  =  ( Base `  H )
)
1511, 14sseqtr4d 3523 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  A
)
1615biantrud 507 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  /\  ( S `  x ) 
C_  A ) ) )
17 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  e.  (SubGrp `  G ) )
18 simplr 754 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  S : dom  S --> (SubGrp `  H )
)
19 eldifi 3608 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( dom  S  \  { x } )  ->  y  e.  dom  S )
2019ad2antll 728 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  y  e.  dom  S )
2118, 20ffvelrnd 6013 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  e.  (SubGrp `  H ) )
229subgss 16071 . . . . . . . . . . . . . . . . 17  |-  ( ( S `  y )  e.  (SubGrp `  H
)  ->  ( S `  y )  C_  ( Base `  H ) )
2321, 22syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  ( Base `  H ) )
2423, 14sseqtr4d 3523 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  A
)
25 eqid 2441 . . . . . . . . . . . . . . . 16  |-  (Cntz `  G )  =  (Cntz `  G )
26 eqid 2441 . . . . . . . . . . . . . . . 16  |-  (Cntz `  H )  =  (Cntz `  H )
2712, 25, 26resscntz 16238 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  (SubGrp `  G )  /\  ( S `  y )  C_  A )  ->  (
(Cntz `  H ) `  ( S `  y
) )  =  ( ( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
2817, 24, 27syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( (Cntz `  H ) `  ( S `  y )
)  =  ( ( (Cntz `  G ) `  ( S `  y
) )  i^i  A
) )
2928sseq2d 3514 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  H
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) ) )
30 ssin 3702 . . . . . . . . . . . . 13  |-  ( ( ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( S `  x )  C_  A
)  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
3129, 30syl6bbr 263 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  H
) `  ( S `  y ) )  <->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  /\  ( S `  x ) 
C_  A ) ) )
3216, 31bitr4d 256 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
(Cntz `  H ) `  ( S `  y
) ) ) )
3332anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  /\  y  e.  ( dom  S  \  { x } ) )  ->  ( ( S `  x )  C_  ( (Cntz `  G
) `  ( S `  y ) )  <->  ( S `  x )  C_  (
(Cntz `  H ) `  ( S `  y
) ) ) )
3433ralbidva 2877 . . . . . . . . 9  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  <->  A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
) ) )
35 subgrcl 16075 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
3635ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  G  e.  Grp )
37 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( Base `  G )  =  (
Base `  G )
3837subgacs 16105 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
39 acsmre 14921 . . . . . . . . . . . . . 14  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4036, 38, 393syl 20 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
4112subggrp 16073 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
4241ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  H  e.  Grp )
439subgacs 16105 . . . . . . . . . . . . . . 15  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
44 acsmre 14921 . . . . . . . . . . . . . . 15  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
4542, 43, 443syl 20 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
46 eqid 2441 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
47 imassrn 5334 . . . . . . . . . . . . . . . . 17  |-  ( S
" ( dom  S  \  { x } ) )  C_  ran  S
48 frn 5723 . . . . . . . . . . . . . . . . . 18  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4948ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ran  S  C_  (SubGrp `  H ) )
5047, 49syl5ss 3497 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  (SubGrp `  H )
)
51 mresspw 14861 . . . . . . . . . . . . . . . . 17  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
5245, 51syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
5350, 52sstrd 3496 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  ~P ( Base `  H
) )
54 sspwuni 4397 . . . . . . . . . . . . . . 15  |-  ( ( S " ( dom 
S  \  { x } ) )  C_  ~P ( Base `  H
)  <->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5553, 54sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5645, 46, 55mrcssidd 14894 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
5746mrccl 14880 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  H )
)  ->  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
5845, 55, 57syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
5912subsubg 16093 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  H
)  <->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) ) )
6059ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  G
)  /\  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
) ) )
6158, 60mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) )
6261simpld 459 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
63 eqid 2441 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6463mrcsscl 14889 . . . . . . . . . . . . 13  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  /\  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
6540, 56, 62, 64syl3anc 1227 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  (
(mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
6613ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  =  (
Base `  H )
)
6755, 66sseqtr4d 3523 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  A )
6837subgss 16071 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
6968ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  C_  ( Base `  G ) )
7067, 69sstrd 3496 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  G
) )
7140, 63, 70mrcssidd 14894 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
7263mrccl 14880 . . . . . . . . . . . . . . 15  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
7340, 70, 72syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G ) )
74 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  e.  (SubGrp `  G ) )
7563mrcsscl 14889 . . . . . . . . . . . . . . 15  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  A  /\  A  e.  (SubGrp `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  A )
7640, 67, 74, 75syl3anc 1227 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
)
7712subsubg 16093 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  H
)  <->  ( ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  G )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) 
C_  A ) ) )
7877ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H )  <->  ( (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  e.  (SubGrp `  G
)  /\  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
) ) )
7973, 76, 78mpbir2and 920 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )
8046mrcsscl 14889 . . . . . . . . . . . . 13  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) )  /\  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  e.  (SubGrp `  H ) )  -> 
( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )
8145, 71, 79, 80syl3anc 1227 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
8265, 81eqssd 3503 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  =  ( (mrCls `  (SubGrp `  H
) ) `  U. ( S " ( dom 
S  \  { x } ) ) ) )
8382ineq2d 3682 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) ) )
84 eqid 2441 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
8512, 84subg0 16076 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
8685ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( 0g `  G )  =  ( 0g `  H ) )
8786sneqd 4022 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  { ( 0g
`  G ) }  =  { ( 0g
`  H ) } )
8883, 87eqeq12d 2463 . . . . . . . . 9  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) }  <->  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )
8934, 88anbi12d 710 . . . . . . . 8  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( ( A. y  e.  ( dom  S 
\  { x }
) ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  <-> 
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) )
9089ralbidva 2877 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  -> 
( A. x  e. 
dom  S ( A. y  e.  ( dom  S 
\  { x }
) ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } )  <->  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) )
9190pm5.32da 641 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
9212subsubg 16093 . . . . . . . . . . . . 13  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) ) )
93 elin 3669 . . . . . . . . . . . . . 14  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A ) )
94 selpw 4000 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~P A  <->  x  C_  A
)
9594anbi2i 694 . . . . . . . . . . . . . 14  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
9693, 95bitri 249 . . . . . . . . . . . . 13  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
9792, 96syl6bbr 263 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  x  e.  ( (SubGrp `  G )  i^i  ~P A ) ) )
9897eqrdv 2438 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  (SubGrp `  H
)  =  ( (SubGrp `  G )  i^i  ~P A ) )
9998sseq2d 3514 . . . . . . . . . 10  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ran 
S  C_  ( (SubGrp `  G )  i^i  ~P A ) ) )
100 ssin 3702 . . . . . . . . . 10  |-  ( ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A )  <->  ran  S  C_  ( (SubGrp `  G )  i^i  ~P A ) )
10199, 100syl6bbr 263 . . . . . . . . 9  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
102101anbi2d 703 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  H
) )  <->  ( S  Fn  dom  S  /\  ( ran  S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) ) )
103 df-f 5578 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  H )
) )
104 df-f 5578 . . . . . . . . . 10  |-  ( S : dom  S --> (SubGrp `  G )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  G )
) )
105104anbi1i 695 . . . . . . . . 9  |-  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S  C_  ~P A
)  <->  ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  G
) )  /\  ran  S 
C_  ~P A ) )
106 anass 649 . . . . . . . . 9  |-  ( ( ( S  Fn  dom  S  /\  ran  S  C_  (SubGrp `  G ) )  /\  ran  S  C_  ~P A )  <->  ( S  Fn  dom  S  /\  ( ran  S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) )
107105, 106bitri 249 . . . . . . . 8  |-  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S  C_  ~P A
)  <->  ( S  Fn  dom  S  /\  ( ran 
S  C_  (SubGrp `  G
)  /\  ran  S  C_  ~P A ) ) )
108102, 103, 1073bitr4g 288 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( S : dom  S --> (SubGrp `  H )  <->  ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
109108anbi1d 704 . . . . . 6  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
11091, 109bitr3d 255 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
111110adantr 465 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
112 dmexg 6712 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
113112adantl 466 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  e.  _V )
114 eqidd 2442 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  =  dom  S )
11541adantr 465 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  H  e.  Grp )
116 eqid 2441 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
11726, 116, 46dmdprd 16898 . . . . . . 7  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( H dom DProd  S  <->  ( H  e. 
Grp  /\  S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
118 3anass 976 . . . . . . 7  |-  ( ( H  e.  Grp  /\  S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) )  <->  ( H  e.  Grp  /\  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
119117, 118syl6bb 261 . . . . . 6  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( H dom DProd  S  <->  ( H  e. 
Grp  /\  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) ) )
120119baibd 907 . . . . 5  |-  ( ( ( dom  S  e. 
_V  /\  dom  S  =  dom  S )  /\  H  e.  Grp )  ->  ( H dom DProd  S  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
121113, 114, 115, 120syl21anc 1226 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( S : dom  S --> (SubGrp `  H )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  H ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  H ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  H ) } ) ) ) )
12235adantr 465 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  G  e.  Grp )
12325, 84, 63dmdprd 16898 . . . . . . . . 9  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
124 3anass 976 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  <->  ( G  e.  Grp  /\  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
125123, 124syl6bb 261 . . . . . . . 8  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) ) )
126125baibd 907 . . . . . . 7  |-  ( ( ( dom  S  e. 
_V  /\  dom  S  =  dom  S )  /\  G  e.  Grp )  ->  ( G dom DProd  S  <->  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
127113, 114, 122, 126syl21anc 1226 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( G dom DProd  S  <->  ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
128127anbi1d 704 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( G dom DProd  S  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  /\  ran  S  C_  ~P A
) ) )
129 an32 796 . . . . 5  |-  ( ( ( S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) )
130128, 129syl6bb 261 . . . 4  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  (
( G dom DProd  S  /\  ran  S  C_  ~P A
)  <->  ( ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A )  /\  A. x  e.  dom  S
( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
131111, 121, 1303bitr4d 285 . . 3  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
132131ex 434 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( S  e.  _V  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) ) )
1333, 6, 132pm5.21ndd 354 1  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093    \ cdif 3455    i^i cin 3457    C_ wss 3458   ~Pcpw 3993   {csn 4010   U.cuni 4230   class class class wbr 4433   dom cdm 4985   ran crn 4986   "cima 4988    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   0gc0g 14709  Moorecmre 14851  mrClscmrc 14852  ACScacs 14854   Grpcgrp 15922  SubGrpcsubg 16064  Cntzccntz 16222   DProd cdprd 16893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-subg 16067  df-cntz 16224  df-dprd 16895
This theorem is referenced by:  subgdprd  16950  ablfaclem3  17006
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