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Theorem dprdres 16515
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 16479 . . . 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 16481 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6 dprdres.3 . . . 4  |-  ( ph  ->  A  C_  I )
7 fssres 5575 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  A  C_  I )  ->  ( S  |`  A ) : A --> (SubGrp `  G )
)
85, 6, 7syl2anc 656 . . 3  |-  ( ph  ->  ( S  |`  A ) : A --> (SubGrp `  G ) )
91ad2antrr 720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  G dom DProd  S )
104ad2antrr 720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  dom  S  =  I )
116ad2antrr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  A  C_  I
)
12 simplr 749 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  A
)
1311, 12sseldd 3354 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  I
)
14 eldifi 3475 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  e.  A
)
1514adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  A
)
1611, 15sseldd 3354 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  I
)
17 eldifsni 3998 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  =/=  x
)
1817adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  =/=  x
)
1918necomd 2693 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  =/=  y
)
20 eqid 2441 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
219, 10, 13, 16, 19, 20dprdcntz 16482 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
22 fvres 5701 . . . . . . . 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 5701 . . . . . . . . 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 5692 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  =  ( (Cntz `  G ) `  ( S `  y )
) )
2721, 23, 263sstr4d 3396 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) ) )
2827ralrimiva 2797 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  ( A  \  {
x } ) ( ( S  |`  A ) `
 x )  C_  ( (Cntz `  G ) `  ( ( S  |`  A ) `  y
) ) )
2922adantl 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
3029ineq1d 3548 . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3231subgacs 15709 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
33 acsmre 14586 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
343, 32, 333syl 20 . . . . . . . . . . 11  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
3534adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
36 eqid 2441 . . . . . . . . . 10  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
37 resss 5131 . . . . . . . . . . . . 13  |-  ( S  |`  A )  C_  S
38 imass1 5200 . . . . . . . . . . . . 13  |-  ( ( S  |`  A )  C_  S  ->  ( ( S  |`  A ) "
( A  \  {
x } ) ) 
C_  ( S "
( A  \  {
x } ) ) )
3937, 38ax-mp 5 . . . . . . . . . . . 12  |-  ( ( S  |`  A ) " ( A  \  { x } ) )  C_  ( S " ( A  \  {
x } ) )
406adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  I )
41 ssdif 3488 . . . . . . . . . . . . 13  |-  ( A 
C_  I  ->  ( A  \  { x }
)  C_  ( I  \  { x } ) )
42 imass2 5201 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  C_  (
I  \  { x } )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4340, 41, 423syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
4439, 43syl5ss 3364 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
45 uniss 4109 . . . . . . . . . . 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 5177 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
48 frn 5562 . . . . . . . . . . . . . . 15  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
495, 48syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
5031subgss 15675 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
51 selpw 3864 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ~P ( Base `  G )  <->  x  C_  ( Base `  G ) )
5250, 51sylibr 212 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (SubGrp `  G
)  ->  x  e.  ~P ( Base `  G
) )
5352ssriv 3357 . . . . . . . . . . . . . . 15  |-  (SubGrp `  G )  C_  ~P ( Base `  G )
5453a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
5549, 54sstrd 3363 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5655adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ran  S 
C_  ~P ( Base `  G
) )
5747, 56syl5ss 3364 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
58 sspwuni 4253 . . . . . . . . . . 11  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
5957, 58sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
6035, 36, 46, 59mrcssd 14558 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) )
61 sslin 3573 . . . . . . . . 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 } ) ) ) ) )
6260, 61syl 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 }
) ) ) ) )
631adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  G dom DProd  S )
644adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  dom  S  =  I )
656sselda 3353 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  I )
66 eqid 2441 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6763, 64, 65, 66, 36dprddisj 16483 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
6862, 67sseqtrd 3389 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
695ffvelrnda 5840 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7065, 69syldan 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7166subg0cl 15682 . . . . . . . . . 10  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  ( S `  x ) )
7270, 71syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( S `  x
) )
7346, 59sstrd 3363 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( Base `  G ) )
7436mrccl 14545 . . . . . . . . . . 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 ) )
7535, 73, 74syl2anc 656 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )
)
7666subg0cl 15682 . . . . . . . . . 10  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )
7775, 76syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) )
7872, 77elind 3537 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7978snssd 4015 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  { ( 0g `  G ) }  C_  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
8068, 79eqssd 3370 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8130, 80eqtrd 2473 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8228, 81jca 529 . . . 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 ) } ) )
8382ralrimiva 2797 . . 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 ) } ) )
84 reldmdprd 16469 . . . . . . . 8  |-  Rel  dom DProd
8584brrelex2i 4876 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
86 dmexg 6508 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
871, 85, 863syl 20 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
884, 87eqeltrrd 2516 . . . . 5  |-  ( ph  ->  I  e.  _V )
8988, 6ssexd 4436 . . . 4  |-  ( ph  ->  A  e.  _V )
90 fdm 5560 . . . . 5  |-  ( ( S  |`  A ) : A --> (SubGrp `  G )  ->  dom  ( S  |`  A )  =  A )
918, 90syl 16 . . . 4  |-  ( ph  ->  dom  ( S  |`  A )  =  A )
9220, 66, 36dmdprd 16470 . . . 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 ) } ) ) ) )
9389, 91, 92syl2anc 656 . . 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 ) } ) ) ) )
943, 8, 83, 93mpbir3and 1166 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  A ) )
95 rnss 5064 . . . . . 6  |-  ( ( S  |`  A )  C_  S  ->  ran  ( S  |`  A )  C_  ran  S )
96 uniss 4109 . . . . . 6  |-  ( ran  ( S  |`  A ) 
C_  ran  S  ->  U.
ran  ( S  |`  A )  C_  U. ran  S )
9737, 95, 96mp2b 10 . . . . 5  |-  U. ran  ( S  |`  A ) 
C_  U. ran  S
9897a1i 11 . . . 4  |-  ( ph  ->  U. ran  ( S  |`  A )  C_  U. ran  S )
99 sspwuni 4253 . . . . 5  |-  ( ran 
S  C_  ~P ( Base `  G )  <->  U. ran  S  C_  ( Base `  G
) )
10055, 99sylib 196 . . . 4  |-  ( ph  ->  U. ran  S  C_  ( Base `  G )
)
10134, 36, 98, 100mrcssd 14558 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
10236dprdspan 16514 . . . 4  |-  ( G dom DProd  ( S  |`  A )  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  ( S  |`  A ) ) )
10394, 102syl 16 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  =  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) ) )
10436dprdspan 16514 . . . 4  |-  ( G dom DProd  S  ->  ( G DProd 
S )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
1051, 104syl 16 . . 3  |-  ( ph  ->  ( G DProd  S )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ran  S
) )
106101, 103, 1053sstr4d 3396 . 2  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) )
10794, 106jca 529 1  |-  ( ph  ->  ( G dom DProd  ( S  |`  A )  /\  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   {csn 3874   U.cuni 4088   class class class wbr 4289   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   0gc0g 14374  Moorecmre 14516  mrClscmrc 14517  ACScacs 14519   Grpcgrp 15406  SubGrpcsubg 15668  Cntzccntz 15826   DProd cdprd 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-gim 15780  df-cntz 15828  df-oppg 15854  df-cmn 16272  df-dprd 16467
This theorem is referenced by:  dprdf1  16520  dprdcntz2  16526  dprddisj2  16527  dprddisj2OLD  16528  dprd2dlem1  16530  dprd2da  16531  dmdprdsplit  16536  dprdsplit  16537  dpjf  16546  dpjidcl  16547  dpjlid  16550  dpjghm  16552  dpjidclOLD  16554  ablfac1eulem  16563  ablfac1eu  16564
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