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Theorem dprdres 16865
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 16829 . . . 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 16831 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
6 dprdres.3 . . . 4  |-  ( ph  ->  A  C_  I )
7 fssres 5749 . . . 4  |-  ( ( S : I --> (SubGrp `  G )  /\  A  C_  I )  ->  ( S  |`  A ) : A --> (SubGrp `  G )
)
85, 6, 7syl2anc 661 . . 3  |-  ( ph  ->  ( S  |`  A ) : A --> (SubGrp `  G ) )
91ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  G dom DProd  S )
104ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  dom  S  =  I )
116ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  A  C_  I
)
12 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  A
)
1311, 12sseldd 3505 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  e.  I
)
14 eldifi 3626 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  e.  A
)
1514adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  A
)
1611, 15sseldd 3505 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  e.  I
)
17 eldifsni 4153 . . . . . . . . . 10  |-  ( y  e.  ( A  \  { x } )  ->  y  =/=  x
)
1817adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  y  =/=  x
)
1918necomd 2738 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  x  =/=  y
)
20 eqid 2467 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
219, 10, 13, 16, 19, 20dprdcntz 16832 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( S `  x )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
22 fvres 5878 . . . . . . . 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 5878 . . . . . . . . 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 5868 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) )  =  ( (Cntz `  G ) `  ( S `  y )
) )
2721, 23, 263sstr4d 3547 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  ( A  \  {
x } ) )  ->  ( ( S  |`  A ) `  x
)  C_  ( (Cntz `  G ) `  (
( S  |`  A ) `
 y ) ) )
2827ralrimiva 2878 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  ( A  \  {
x } ) ( ( S  |`  A ) `
 x )  C_  ( (Cntz `  G ) `  ( ( S  |`  A ) `  y
) ) )
2922adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A ) `
 x )  =  ( S `  x
) )
3029ineq1d 3699 . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3231subgacs 16031 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
33 acsmre 14903 . . . . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
36 eqid 2467 . . . . . . . . . 10  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
37 resss 5295 . . . . . . . . . . . . 13  |-  ( S  |`  A )  C_  S
38 imass1 5369 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  I )
41 ssdif 3639 . . . . . . . . . . . . 13  |-  ( A 
C_  I  ->  ( A  \  { x }
)  C_  ( I  \  { x } ) )
42 imass2 5370 . . . . . . . . . . . . 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 3515 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( S " ( I  \  {
x } ) ) )
45 uniss 4266 . . . . . . . . . . 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 5346 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
48 frn 5735 . . . . . . . . . . . . . . 15  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
495, 48syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
5031subgss 15997 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
51 selpw 4017 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ~P ( Base `  G )  <->  x  C_  ( Base `  G ) )
5250, 51sylibr 212 . . . . . . . . . . . . . . . 16  |-  ( x  e.  (SubGrp `  G
)  ->  x  e.  ~P ( Base `  G
) )
5352ssriv 3508 . . . . . . . . . . . . . . 15  |-  (SubGrp `  G )  C_  ~P ( Base `  G )
5453a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
5549, 54sstrd 3514 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5655adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ran  S 
C_  ~P ( Base `  G
) )
5747, 56syl5ss 3515 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
58 sspwuni 4411 . . . . . . . . . . 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 14875 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) )
61 sslin 3724 . . . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  G dom DProd  S )
644adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  dom  S  =  I )
656sselda 3504 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  I )
66 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6763, 64, 65, 66, 36dprddisj 16833 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
6862, 67sseqtrd 3540 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
695ffvelrnda 6019 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7065, 69syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  e.  (SubGrp `  G )
)
7166subg0cl 16004 . . . . . . . . . 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 3514 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  U. (
( S  |`  A )
" ( A  \  { x } ) )  C_  ( Base `  G ) )
7436mrccl 14862 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) )  e.  (SubGrp `  G )
)
7666subg0cl 16004 . . . . . . . . . 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 3688 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( 0g `  G )  e.  ( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
7978snssd 4172 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  { ( 0g `  G ) }  C_  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( S  |`  A ) " ( A  \  { x }
) ) ) ) )
8068, 79eqssd 3521 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8130, 80eqtrd 2508 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( S  |`  A ) `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( S  |`  A ) " ( A  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8228, 81jca 532 . . . 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 2878 . . 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 16819 . . . . . . . 8  |-  Rel  dom DProd
8584brrelex2i 5040 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
86 dmexg 6712 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
871, 85, 863syl 20 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
884, 87eqeltrrd 2556 . . . . 5  |-  ( ph  ->  I  e.  _V )
8988, 6ssexd 4594 . . . 4  |-  ( ph  ->  A  e.  _V )
90 fdm 5733 . . . . 5  |-  ( ( S  |`  A ) : A --> (SubGrp `  G )  ->  dom  ( S  |`  A )  =  A )
918, 90syl 16 . . . 4  |-  ( ph  ->  dom  ( S  |`  A )  =  A )
9220, 66, 36dmdprd 16820 . . . 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 661 . . 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 1179 . 2  |-  ( ph  ->  G dom DProd  ( S  |`  A ) )
95 rnss 5229 . . . . . 6  |-  ( ( S  |`  A )  C_  S  ->  ran  ( S  |`  A )  C_  ran  S )
96 uniss 4266 . . . . . 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 4411 . . . . 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 14875 . . 3  |-  ( ph  ->  ( (mrCls `  (SubGrp `  G ) ) `  U. ran  ( S  |`  A ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  S ) )
10236dprdspan 16864 . . . 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 16864 . . . 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 3547 . 2  |-  ( ph  ->  ( G DProd  ( S  |`  A ) )  C_  ( G DProd  S ) )
10794, 106jca 532 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   0gc0g 14691  Moorecmre 14833  mrClscmrc 14834  ACScacs 14836   Grpcgrp 15723  SubGrpcsubg 15990  Cntzccntz 16148   DProd cdprd 16815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-gim 16102  df-cntz 16150  df-oppg 16176  df-cmn 16596  df-dprd 16817
This theorem is referenced by:  dprdf1  16870  dprdcntz2  16876  dprddisj2  16877  dprddisj2OLD  16878  dprd2dlem1  16880  dprd2da  16881  dmdprdsplit  16886  dprdsplit  16887  dpjf  16896  dpjidcl  16897  dpjlid  16900  dpjghm  16902  dpjidclOLD  16904  ablfac1eulem  16913  ablfac1eu  16914
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