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Theorem dprdres 15578
Description: Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdres.1  |-  ( ph  ->  G dom DProd  S )
dprdres.2  |-  ( ph  ->  dom  S  =  I )
dprdres.3  |-  ( ph  ->  A  C_  I )
Assertion
Ref Expression
dprdres  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  /\  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) ) )

Proof of Theorem dprdres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdres.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
2 dprdgrp 15555 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
4 dprdres.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
51, 4dprdf2 15557 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6 dprdres.3 . . . 4  |-  ( ph  ->  A  C_  I )
7 fssres 5602 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  A  C_  I )  ->  ( S  |`  A ) : A --> (SubGrp `  G )
)
85, 6, 7syl2anc 643 . . 3  |-  ( ph  ->  ( S  |`  A ) : A --> (SubGrp `  G ) )
91ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  G dom DProd  S )
104ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  dom  S  =  I )
116ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  A  C_  I
)
12 simplr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  A
)
1311, 12sseldd 3341 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  I
)
14 eldifi 3461 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  e.  A
)
1514adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  A
)
1611, 15sseldd 3341 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  I
)
17 eldifsni 3920 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  =/=  x
)
1817adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  =/=  x
)
1918necomd 2681 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  =/=  y
)
20 eqid 2435 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
219, 10, 13, 16, 19, 20dprdcntz 15558 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
22 fvres 5737 . . . . . . . 8  |-  ( x  e.  A  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
2312, 22syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  =  ( S `
 x ) )
24 fvres 5737 . . . . . . . . 9  |-  ( y  e.  A  ->  (
( S  |`  A ) `
 y )  =  ( S `  y
) )
2515, 24syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  y
)  =  ( S `
 y ) )
2625fveq2d 5724 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  =  ( (Cntz `  G ) `  ( S `  y )
) )
2721, 23, 263sstr4d 3383 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) ) )
2827ralrimiva 2781 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  ( A  \  {
x } ) ( ( S  |`  A ) `
 x )  C_  ( (Cntz `  G ) `  ( ( S  |`  A ) `  y
) ) )
2922adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
3029ineq1d 3533 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
31 eqid 2435 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3231subgacs 14967 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
33 acsmre 13869 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
343, 32, 333syl 19 . . . . . . . . . . 11  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
3534adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
36 eqid 2435 . . . . . . . . . 10  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
37 resss 5162 . . . . . . . . . . . . 13  |-  ( S  |`  A )  C_  S
38 imass1 5231 . . . . . . . . . . . . 13  |-  ( ( S  |`  A )  C_  S  ->  ( ( S  |`  A ) "
( A  \  {
x } ) ) 
C_  ( S "
( A  \  {
x } ) ) )
3937, 38ax-mp 8 . . . . . . . . . . . 12  |-  ( ( S  |`  A ) " ( A  \  { x } ) )  C_  ( S " ( A  \  {
x } ) )
406adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  I )
41 ssdif 3474 . . . . . . . . . . . . 13  |-  ( A 
C_  I  ->  ( A  \  { x }
)  C_  ( I  \  { x } ) )
42 imass2 5232 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  C_  (
I  \  { x } )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4340, 41, 423syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4439, 43syl5ss 3351 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
45 uniss 4028 . . . . . . . . . . 11  |-  ( ( ( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) )  ->  U. ( ( S  |`  A ) " ( A  \  { x }
) )  C_  U. ( S " ( I  \  { x } ) ) )
4644, 45syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  U. ( S " ( I  \  { x } ) ) )
47 imassrn 5208 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
48 frn 5589 . . . . . . . . . . . . . . 15  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
495, 48syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
5031subgss 14937 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
51 vex 2951 . . . . . . . . . . . . . . . . . 18  |-  x  e. 
_V
5251elpw 3797 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ~P ( Base `  G )  <->  x  C_  ( Base `  G ) )
5350, 52sylibr 204 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (SubGrp `  G
)  ->  x  e.  ~P ( Base `  G
) )
5453ssriv 3344 . . . . . . . . . . . . . . 15  |-  (SubGrp `  G )  C_  ~P ( Base `  G )
5554a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
5649, 55sstrd 3350 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5756adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ran  S 
C_  ~P ( Base `  G
) )
5847, 57syl5ss 3351 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
59 sspwuni 4168 . . . . . . . . . . 11  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
6058, 59sylib 189 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
6135, 36, 46, 60mrcssd 13841 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) )
62 sslin 3559 . . . . . . . . 9  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) 
C_  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) ) )
6361, 62syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) )
641adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  G dom DProd  S )
654adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  dom  S  =  I )
666sselda 3340 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  I )
67 eqid 2435 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6864, 65, 66, 67, 36dprddisj 15559 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
6963, 68sseqtrd 3376 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
705ffvelrnda 5862 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7166, 70syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7267subg0cl 14944 . . . . . . . . . 10  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  ( S `  x ) )
7371, 72syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( S `  x
) )
7446, 60sstrd 3350 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( Base `  G ) )
7536mrccl 13828 . . . . . . . . . . 11  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( ( S  |`  A ) " ( A  \  { x }
) )  C_  ( Base `  G ) )  ->  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( ( S  |`  A ) " ( A  \  { x } ) ) )  e.  (SubGrp `  G ) )
7635, 74, 75syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )
)
7767subg0cl 14944 . . . . . . . . . 10  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )
7876, 77syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )
79 elin 3522 . . . . . . . . 9  |-  ( ( 0g `  G )  e.  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )  <-> 
( ( 0g `  G )  e.  ( S `  x )  /\  ( 0g `  G )  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
8073, 78, 79sylanbrc 646 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
8180snssd 3935 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  { ( 0g `  G ) }  C_  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
8269, 81eqssd 3357 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8330, 82eqtrd 2467 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8428, 83jca 519 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  ( A  \  { x }
) ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  /\  ( ( ( S  |`  A ) `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )  =  { ( 0g
`  G ) } ) )
8584ralrimiva 2781 . . 3  |-  ( ph  ->  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) )
86 reldmdprd 15550 . . . . . . . 8  |-  Rel  dom DProd
8786brrelex2i 4911 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
88 dmexg 5122 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
891, 87, 883syl 19 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
904, 89eqeltrrd 2510 . . . . 5  |-  ( ph  ->  I  e.  _V )
9190, 6ssexd 4342 . . . 4  |-  ( ph  ->  A  e.  _V )
92 fdm 5587 . . . . 5  |-  ( ( S  |`  A ) : A --> (SubGrp `  G )  ->  dom  ( S  |`  A )  =  A )
938, 92syl 16 . . . 4  |-  ( ph  ->  dom  ( S  |`  A )  =  A )
9420, 67, 36dmdprd 15551 . . . 4  |-  ( ( A  e.  _V  /\  dom  ( S  |`  A )  =  A )  -> 
( G dom DProd  ( S  |`  A )  <->  ( G  e.  Grp  /\  ( S  |`  A ) : A --> (SubGrp `  G )  /\  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
9591, 93, 94syl2anc 643 . . 3  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  <->  ( G  e.  Grp  /\  ( S  |`  A ) : A --> (SubGrp `  G )  /\  A. x  e.  A  ( A. y  e.  ( A  \  { x } ) ( ( S  |`  A ) `  x )  C_  (
(Cntz `  G ) `  ( ( S  |`  A ) `  y
) )  /\  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
963, 8, 85, 95mpbir3and 1137 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  A ) )
97 rnss 5090 . . . . . 6  |-  ( ( S  |`  A )  C_  S  ->  ran  ( S  |`  A )  C_  ran  S )
98 uniss 4028 . . . . . 6  |-  ( ran  ( S  |`  A ) 
C_  ran  S  ->  U.
ran  ( S  |`  A )  C_  U. ran  S )
9937, 97, 98mp2b 10 . . . . 5  |-  U. ran  ( S  |`  A ) 
C_  U. ran  S
10099a1i 11 . . . 4  |-  ( ph  ->  U. ran  ( S  |`  A )  C_  U. ran  S )
101 sspwuni 4168 . . . . 5  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
10256, 101sylib 189 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
10334, 36, 100, 102mrcssd 13841 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
10436dprdspan 15577 . . . 4  |-  ( G dom DProd  ( S  |`  A )  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  ( S  |`  A ) ) )
10596, 104syl 16 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) ) )
10636dprdspan 15577 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
1071, 106syl 16 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
108103, 105, 1073sstr4d 3383 . 2  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) )
10996, 108jca 519 1  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  /\  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   0gc0g 13715  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Grpcgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546
This theorem is referenced by:  dprdf1  15583  dprdcntz2  15588  dprddisj2  15589  dprd2dlem1  15591  dprd2da  15592  dmdprdsplit  15597  dprdsplit  15598  dpjf  15607  dpjidcl  15608  dpjlid  15611  dpjghm  15613  ablfac1eulem  15622  ablfac1eu  15623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-gim 15038  df-cntz 15108  df-oppg 15134  df-cmn 15406  df-dprd 15548
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