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Theorem resscatc 15306
Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat `  U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resscatc.c  |-  C  =  (CatCat `  U )
resscatc.d  |-  D  =  (CatCat `  V )
resscatc.1  |-  ( ph  ->  U  e.  W )
resscatc.2  |-  ( ph  ->  V  C_  U )
Assertion
Ref Expression
resscatc  |-  ( ph  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )

Proof of Theorem resscatc
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resscatc.d . . . . . 6  |-  D  =  (CatCat `  V )
2 eqid 2443 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
4 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
53, 4ssexd 4584 . . . . . . 7  |-  ( ph  ->  V  e.  _V )
65adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  V  e.  _V )
7 eqid 2443 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
8 simprl 756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( V  i^i  Cat ) )
91, 2, 5catcbas 15298 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  ( V  i^i  Cat ) )
109adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( Base `  D )  =  ( V  i^i  Cat ) )
118, 10eleqtrrd 2534 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  D
) )
12 simprr 757 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( V  i^i  Cat ) )
1312, 10eleqtrrd 2534 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
141, 2, 6, 7, 11, 13catchom 15300 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  D
) y )  =  ( x  Func  y
) )
15 resscatc.c . . . . . 6  |-  C  =  (CatCat `  U )
16 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
173adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  U  e.  W )
18 eqid 2443 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
1915, 16, 3catcbas 15298 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2019ineq2d 3685 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
21 inass 3693 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2220, 21syl6reqr 2503 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
23 df-ss 3475 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
244, 23sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2524ineq1d 3684 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
26 eqid 2443 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2726, 16ressbas 14564 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  ( V  i^i  ( Base `  C
) )  =  (
Base `  ( Cs  V
) ) )
285, 27syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( Base `  ( Cs  V ) ) )
2922, 25, 283eqtr3d 2492 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3026, 16ressbasss 14566 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3129, 30syl6eqss 3539 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  Cat )  C_  ( Base `  C
) )
3231adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( V  i^i  Cat )  C_  ( Base `  C )
)
3332, 8sseldd 3490 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3432, 12sseldd 3490 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3515, 16, 17, 18, 33, 34catchom 15300 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x  Func  y
) )
3626, 18resshom 14693 . . . . . . 7  |-  ( V  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  ( Cs  V ) ) )
375, 36syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  ( Cs  V ) ) )
3837oveqdr 6305 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x ( Hom  `  ( Cs  V ) ) y ) )
3914, 35, 383eqtr2rd 2491 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) )
4039ralrimivva 2864 . . 3  |-  ( ph  ->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) ( x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) )
41 eqid 2443 . . . 4  |-  ( Hom  `  ( Cs  V ) )  =  ( Hom  `  ( Cs  V ) )
429eqcomd 2451 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4341, 7, 29, 42homfeq 14966 . . 3  |-  ( ph  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  <->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat )
( x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) ) )
4440, 43mpbird 232 . 2  |-  ( ph  ->  ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D ) )
455ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  V  e.  _V )
46 eqid 2443 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
47 simplr1 1039 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  x  e.  ( V  i^i  Cat )
)
489ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( Base `  D
)  =  ( V  i^i  Cat ) )
4947, 48eleqtrrd 2534 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  x  e.  (
Base `  D )
)
50 simplr2 1040 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  y  e.  ( V  i^i  Cat )
)
5150, 48eleqtrrd 2534 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  y  e.  (
Base `  D )
)
52 simplr3 1041 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  z  e.  ( V  i^i  Cat )
)
5352, 48eleqtrrd 2534 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  z  e.  (
Base `  D )
)
54 simprl 756 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  f  e.  ( x ( Hom  `  D
) y ) )
551, 2, 45, 7, 49, 51catchom 15300 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( x ( Hom  `  D )
y )  =  ( x  Func  y )
)
5654, 55eleqtrd 2533 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  f  e.  ( x  Func  y )
)
57 simprr 757 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  g  e.  ( y ( Hom  `  D
) z ) )
581, 2, 45, 7, 51, 53catchom 15300 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( y ( Hom  `  D )
z )  =  ( y  Func  z )
)
5957, 58eleqtrd 2533 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  g  e.  ( y  Func  z )
)
601, 2, 45, 46, 49, 51, 53, 56, 59catcco 15302 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  D )
z ) f )  =  ( g  o.func  f ) )
613ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  U  e.  W
)
62 eqid 2443 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
6331ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( V  i^i  Cat )  C_  ( Base `  C ) )
6463, 47sseldd 3490 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  x  e.  (
Base `  C )
)
6563, 50sseldd 3490 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  y  e.  (
Base `  C )
)
6663, 52sseldd 3490 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  z  e.  (
Base `  C )
)
6715, 16, 61, 62, 64, 65, 66, 56, 59catcco 15302 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  =  ( g  o.func  f ) )
6826, 62ressco 14694 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  (comp `  C )  =  (comp `  ( Cs  V ) ) )
695, 68syl 16 . . . . . . . . . 10  |-  ( ph  ->  (comp `  C )  =  (comp `  ( Cs  V
) ) )
7069ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  (comp `  C
)  =  (comp `  ( Cs  V ) ) )
7170oveqd 6298 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( <. x ,  y >. (comp `  C ) z )  =  ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) )
7271oveqd 6298 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) )
7360, 67, 723eqtr2d 2490 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x ( Hom  `  D
) y )  /\  g  e.  ( y
( Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  D )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) )
7473ralrimivva 2864 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  ->  A. f  e.  ( x ( Hom  `  D ) y ) A. g  e.  ( y ( Hom  `  D
) z ) ( g ( <. x ,  y >. (comp `  D ) z ) f )  =  ( g ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) f ) )
7574ralrimivvva 2865 . . . 4  |-  ( ph  ->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) A. z  e.  ( V  i^i  Cat ) A. f  e.  ( x ( Hom  `  D ) y ) A. g  e.  ( y ( Hom  `  D
) z ) ( g ( <. x ,  y >. (comp `  D ) z ) f )  =  ( g ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) f ) )
76 eqid 2443 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7744eqcomd 2451 . . . . 5  |-  ( ph  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  V ) ) )
7846, 76, 7, 42, 29, 77comfeq 14978 . . . 4  |-  ( ph  ->  ( (compf `  D )  =  (compf `  ( Cs  V ) )  <->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) A. z  e.  ( V  i^i  Cat ) A. f  e.  ( x
( Hom  `  D ) y ) A. g  e.  ( y ( Hom  `  D ) z ) ( g ( <.
x ,  y >.
(comp `  D )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) ) )
7975, 78mpbird 232 . . 3  |-  ( ph  ->  (compf `  D )  =  (compf `  ( Cs  V ) ) )
8079eqcomd 2451 . 2  |-  ( ph  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
8144, 80jca 532 1  |-  ( ph  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   <.cop 4020   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   Hom chom 14585  compcco 14586   Catccat 14938   Hom f chomf 14940  compfccomf 14941    Func cfunc 15097    o.func ccofu 15099  CatCatccatc 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-hom 14598  df-cco 14599  df-homf 14944  df-comf 14945  df-catc 15296
This theorem is referenced by: (None)
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