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Theorem subgdmdprd 15479
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 15445 . . . 4  |-  Rel  dom DProd
21brrelex2i 4833 . . 3  |-  ( H dom DProd  S  ->  S  e. 
_V )
32a1i 10 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  ->  S  e.  _V ) )
41brrelex2i 4833 . . . 4  |-  ( G dom DProd  S  ->  S  e. 
_V )
54adantr 451 . . 3  |-  ( ( G dom DProd  S  /\  ran  S  C_  ~P A
)  ->  S  e.  _V )
65a1i 10 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( ( G dom DProd  S  /\  ran  S  C_ 
~P A )  ->  S  e.  _V )
)
7 ffvelrn 5770 . . . . . . . . . . . . . . . 16  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  x  e.  dom  S )  ->  ( S `  x )  e.  (SubGrp `  H ) )
87ad2ant2lr 728 . . . . . . . . . . . . . . 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 2366 . . . . . . . . . . . . . . . 16  |-  ( Base `  H )  =  (
Base `  H )
109subgss 14832 . . . . . . . . . . . . . . 15  |-  ( ( S `  x )  e.  (SubGrp `  H
)  ->  ( S `  x )  C_  ( Base `  H ) )
118, 10syl 15 . . . . . . . . . . . . . 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 14835 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1413ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  A  =  ( Base `  H )
)
1511, 14sseqtr4d 3301 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  x )  C_  A
)
1615biantrud 493 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . 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 3385 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( dom  S  \  { x } )  ->  y  e.  dom  S )
2019ad2antll 709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  y  e.  dom  S )
21 ffvelrn 5770 . . . . . . . . . . . . . . . . . 18  |-  ( ( S : dom  S --> (SubGrp `  H )  /\  y  e.  dom  S )  ->  ( S `  y )  e.  (SubGrp `  H ) )
2218, 20, 21syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  e.  (SubGrp `  H ) )
239subgss 14832 . . . . . . . . . . . . . . . . 17  |-  ( ( S `  y )  e.  (SubGrp `  H
)  ->  ( S `  y )  C_  ( Base `  H ) )
2422, 23syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  ( Base `  H ) )
2524, 14sseqtr4d 3301 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  ( x  e.  dom  S  /\  y  e.  ( dom  S  \  {
x } ) ) )  ->  ( S `  y )  C_  A
)
26 eqid 2366 . . . . . . . . . . . . . . . 16  |-  (Cntz `  G )  =  (Cntz `  G )
27 eqid 2366 . . . . . . . . . . . . . . . 16  |-  (Cntz `  H )  =  (Cntz `  H )
2812, 26, 27resscntz 15017 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  (SubGrp `  G )  /\  ( S `  y )  C_  A )  ->  (
(Cntz `  H ) `  ( S `  y
) )  =  ( ( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
2917, 25, 28syl2anc 642 . . . . . . . . . . . . . 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
) )
3029sseq2d 3292 . . . . . . . . . . . . 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 ) ) )
31 ssin 3479 . . . . . . . . . . . . 13  |-  ( ( ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( S `  x )  C_  A
)  <->  ( S `  x )  C_  (
( (Cntz `  G
) `  ( S `  y ) )  i^i 
A ) )
3230, 31syl6bbr 254 . . . . . . . . . . . 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 ) ) )
3316, 32bitr4d 247 . . . . . . . . . . 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
) ) ) )
3433anassrs 629 . . . . . . . . . 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
) ) ) )
3534ralbidva 2644 . . . . . . . . 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 )
) ) )
36 subgrcl 14836 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubGrp `  G
)  ->  G  e.  Grp )
3736ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  G  e.  Grp )
38 eqid 2366 . . . . . . . . . . . . . . 15  |-  ( Base `  G )  =  (
Base `  G )
3938subgacs 14862 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
40 acsmre 13764 . . . . . . . . . . . . . 14  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4137, 39, 403syl 18 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
4212subggrp 14834 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
4342ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  H  e.  Grp )
449subgacs 14862 . . . . . . . . . . . . . . 15  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
45 acsmre 13764 . . . . . . . . . . . . . . 15  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
4643, 44, 453syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
47 imassrn 5128 . . . . . . . . . . . . . . . . 17  |-  ( S
" ( dom  S  \  { x } ) )  C_  ran  S
48 frn 5501 . . . . . . . . . . . . . . . . . 18  |-  ( S : dom  S --> (SubGrp `  H )  ->  ran  S 
C_  (SubGrp `  H )
)
4948ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ran  S  C_  (SubGrp `  H ) )
5047, 49syl5ss 3276 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  (SubGrp `  H )
)
51 mresspw 13704 . . . . . . . . . . . . . . . . 17  |-  ( (SubGrp `  H )  e.  (Moore `  ( Base `  H
) )  ->  (SubGrp `  H )  C_  ~P ( Base `  H )
)
5246, 51syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  (SubGrp `  H )  C_ 
~P ( Base `  H
) )
5350, 52sstrd 3275 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( S "
( dom  S  \  {
x } ) ) 
C_  ~P ( Base `  H
) )
54 sspwuni 4089 . . . . . . . . . . . . . . 15  |-  ( ( S " ( dom 
S  \  { x } ) )  C_  ~P ( Base `  H
)  <->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
5553, 54sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  H
) )
56 eqid 2366 . . . . . . . . . . . . . . 15  |-  (mrCls `  (SubGrp `  H ) )  =  (mrCls `  (SubGrp `  H ) )
5756mrcssid 13729 . . . . . . . . . . . . . 14  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  H )
)  ->  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  H )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )
5846, 55, 57syl2anc 642 . . . . . . . . . . . . 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 } ) ) ) )
5956mrccl 13723 . . . . . . . . . . . . . . . 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 ) )
6046, 55, 59syl2anc 642 . . . . . . . . . . . . . . 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 ) )
6112subsubg 14850 . . . . . . . . . . . . . . . 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 ) ) )
6261ad2antrr 706 . . . . . . . . . . . . . . 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
) ) )
6360, 62mpbid 201 . . . . . . . . . . . . . 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 ) )
6463simpld 445 . . . . . . . . . . . . 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 ) )
65 eqid 2366 . . . . . . . . . . . . . 14  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
6665mrcsscl 13732 . . . . . . . . . . . . 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 } ) ) ) )
6741, 58, 64, 66syl3anc 1183 . . . . . . . . . . . 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 } ) ) ) )
6813ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  =  (
Base `  H )
)
6955, 68sseqtr4d 3301 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  A )
7038subgss 14832 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
7170ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  C_  ( Base `  G ) )
7269, 71sstrd 3275 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  U. ( S "
( dom  S  \  {
x } ) ) 
C_  ( Base `  G
) )
7365mrcssid 13729 . . . . . . . . . . . . . 14  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( dom  S  \  { x } ) )  C_  ( Base `  G )
)  ->  U. ( S " ( dom  S  \  { x } ) )  C_  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( dom  S  \  { x } ) ) ) )
7441, 72, 73syl2anc 642 . . . . . . . . . . . . 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 } ) ) ) )
7565mrccl 13723 . . . . . . . . . . . . . . 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 ) )
7641, 72, 75syl2anc 642 . . . . . . . . . . . . . 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 ) )
77 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  A  e.  (SubGrp `  G ) )
7865mrcsscl 13732 . . . . . . . . . . . . . . 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 )
7941, 69, 77, 78syl3anc 1183 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( S
" ( dom  S  \  { x } ) ) )  C_  A
)
8012subsubg 14850 . . . . . . . . . . . . . . 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 ) ) )
8180ad2antrr 706 . . . . . . . . . . . . . 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
) ) )
8276, 79, 81mpbir2and 888 . . . . . . . . . . . . 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 ) )
8356mrcsscl 13732 . . . . . . . . . . . . 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 } ) ) ) )
8446, 74, 82, 83syl3anc 1183 . . . . . . . . . . . 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 } ) ) ) )
8567, 84eqssd 3282 . . . . . . . . . . 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 } ) ) ) )
8685ineq2d 3458 . . . . . . . . . 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 } ) ) ) ) )
87 eqid 2366 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
8812, 87subg0 14837 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
8988ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  ( 0g `  G )  =  ( 0g `  H ) )
9089sneqd 3742 . . . . . . . . . 10  |-  ( ( ( A  e.  (SubGrp `  G )  /\  S : dom  S --> (SubGrp `  H ) )  /\  x  e.  dom  S )  ->  { ( 0g
`  G ) }  =  { ( 0g
`  H ) } )
9186, 90eqeq12d 2380 . . . . . . . . 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 ) } ) )
9235, 91anbi12d 691 . . . . . . . 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 ) } ) ) )
9392ralbidva 2644 . . . . . . 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 ) } ) ) )
9493pm5.32da 622 . . . . . 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 ) } ) ) ) )
9512subsubg 14850 . . . . . . . . . . . . 13  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) ) )
96 elin 3446 . . . . . . . . . . . . . 14  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A ) )
97 vex 2876 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
9897elpw 3720 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~P A  <->  x  C_  A
)
9998anbi2i 675 . . . . . . . . . . . . . 14  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
10096, 99bitri 240 . . . . . . . . . . . . 13  |-  ( x  e.  ( (SubGrp `  G )  i^i  ~P A )  <->  ( x  e.  (SubGrp `  G )  /\  x  C_  A ) )
10195, 100syl6bbr 254 . . . . . . . . . . . 12  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  (SubGrp `  H )  <->  x  e.  ( (SubGrp `  G )  i^i  ~P A ) ) )
102101eqrdv 2364 . . . . . . . . . . 11  |-  ( A  e.  (SubGrp `  G
)  ->  (SubGrp `  H
)  =  ( (SubGrp `  G )  i^i  ~P A ) )
103102sseq2d 3292 . . . . . . . . . 10  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ran 
S  C_  ( (SubGrp `  G )  i^i  ~P A ) ) )
104 ssin 3479 . . . . . . . . . 10  |-  ( ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A )  <->  ran  S  C_  ( (SubGrp `  G )  i^i  ~P A ) )
105103, 104syl6bbr 254 . . . . . . . . 9  |-  ( A  e.  (SubGrp `  G
)  ->  ( ran  S 
C_  (SubGrp `  H )  <->  ( ran  S  C_  (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
106105anbi2d 684 . . . . . . . 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 ) ) ) )
107 df-f 5362 . . . . . . . 8  |-  ( S : dom  S --> (SubGrp `  H )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  H )
) )
108 df-f 5362 . . . . . . . . . 10  |-  ( S : dom  S --> (SubGrp `  G )  <->  ( S  Fn  dom  S  /\  ran  S 
C_  (SubGrp `  G )
) )
109108anbi1i 676 . . . . . . . . 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 ) )
110 anass 630 . . . . . . . . 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 ) ) )
111109, 110bitri 240 . . . . . . . 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 ) ) )
112106, 107, 1113bitr4g 279 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( S : dom  S --> (SubGrp `  H )  <->  ( S : dom  S --> (SubGrp `  G )  /\  ran  S 
C_  ~P A ) ) )
113112anbi1d 685 . . . . . 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 ) } ) ) ) )
11494, 113bitr3d 246 . . . . 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 ) } ) ) ) )
115114adantr 451 . . . 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 ) } ) ) ) )
116 dmexg 5042 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
117116adantl 452 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  e.  _V )
118 eqidd 2367 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  dom  S  =  dom  S )
11942adantr 451 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  H  e.  Grp )
120 eqid 2366 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
12127, 120, 56dmdprd 15446 . . . . . . 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 ) } ) ) ) )
122 3anass 939 . . . . . . 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 ) } ) ) ) )
123121, 122syl6bb 252 . . . . . 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 ) } ) ) ) ) )
124123baibd 875 . . . . 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 ) } ) ) ) )
125117, 118, 119, 124syl21anc 1182 . . . 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 ) } ) ) ) )
12636adantr 451 . . . . . . 7  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  G  e.  Grp )
12726, 87, 65dmdprd 15446 . . . . . . . . 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 ) } ) ) ) )
128 3anass 939 . . . . . . . . 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 ) } ) ) ) )
129127, 128syl6bb 252 . . . . . . . 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 ) } ) ) ) ) )
130129baibd 875 . . . . . . 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 ) } ) ) ) )
131117, 118, 126, 130syl21anc 1182 . . . . . 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 ) } ) ) ) )
132131anbi1d 685 . . . . 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
) ) )
133 an32 773 . . . . 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 ) } ) ) )
134132, 133syl6bb 252 . . . 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 ) } ) ) ) )
135115, 125, 1343bitr4d 276 . . 3  |-  ( ( A  e.  (SubGrp `  G )  /\  S  e.  _V )  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
136135ex 423 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( S  e.  _V  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) ) )
1373, 6, 136pm5.21ndd 343 1  |-  ( A  e.  (SubGrp `  G
)  ->  ( H dom DProd  S  <->  ( G dom DProd  S  /\  ran  S  C_  ~P A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873    \ cdif 3235    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   {csn 3729   U.cuni 3929   class class class wbr 4125   dom cdm 4792   ran crn 4793   "cima 4795    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   Basecbs 13356   ↾s cress 13357   0gc0g 13610  Moorecmre 13694  mrClscmrc 13695  ACScacs 13697   Grpcgrp 14572  SubGrpcsubg 14825  Cntzccntz 15001   DProd cdprd 15441
This theorem is referenced by:  subgdprd  15480  ablfaclem3  15532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-0g 13614  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-subg 14828  df-cntz 15003  df-dprd 15443
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