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Theorem resscatc 15279
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 2460 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
4 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
53, 4ssexd 4587 . . . . . . 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 2460 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
8 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( V  i^i  Cat ) )
91, 2, 5catcbas 15271 . . . . . . . 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 2551 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  D
) )
12 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( V  i^i  Cat ) )
1312, 10eleqtrrd 2551 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
141, 2, 6, 7, 11, 13catchom 15273 . . . . 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 2460 . . . . . 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 2460 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
1915, 16, 3catcbas 15271 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2019ineq2d 3693 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
21 inass 3701 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2220, 21syl6reqr 2520 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
23 df-ss 3483 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
244, 23sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2524ineq1d 3692 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
26 eqid 2460 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2726, 16ressbas 14534 . . . . . . . . . . 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 2509 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3026, 16ressbasss 14536 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3129, 30syl6eqss 3547 . . . . . . . 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 3498 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3432, 12sseldd 3498 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3515, 16, 17, 18, 33, 34catchom 15273 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x  Func  y
) )
3626, 18resshom 14663 . . . . . . 7  |-  ( V  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  ( Cs  V ) ) )
375, 36syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  ( Cs  V ) ) )
3837proplem3 14935 . . . . 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 2508 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) )
4039ralrimivva 2878 . . 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 2460 . . . 4  |-  ( Hom  `  ( Cs  V ) )  =  ( Hom  `  ( Cs  V ) )
429eqcomd 2468 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4341, 7, 29, 42homfeq 14939 . . 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 2460 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
47 simplr1 1033 . . . . . . . . 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 2551 . . . . . . . 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 1034 . . . . . . . . 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 2551 . . . . . . . 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 1035 . . . . . . . . 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 2551 . . . . . . . 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 755 . . . . . . . . 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 15273 . . . . . . . . 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 2550 . . . . . . . 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 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 ) ) )  ->  g  e.  ( y ( Hom  `  D
) z ) )
581, 2, 45, 7, 51, 53catchom 15273 . . . . . . . . 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 2550 . . . . . . . 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 15275 . . . . . . 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 2460 . . . . . . . 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 3498 . . . . . . . 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 3498 . . . . . . . 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 3498 . . . . . . . 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 15275 . . . . . . 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 14664 . . . . . . . . . . 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 6292 . . . . . . . 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 6292 . . . . . . 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 2507 . . . . . 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 2878 . . . . 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 2879 . . . 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 2460 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7744eqcomd 2468 . . . . 5  |-  ( ph  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  V ) ) )
7846, 76, 7, 42, 29, 77comfeq 14951 . . . 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 2468 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    i^i cin 3468    C_ wss 3469   <.cop 4026   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   Hom chom 14555  compcco 14556   Catccat 14908   Hom f chomf 14910  compfccomf 14911    Func cfunc 15070    o.func ccofu 15072  CatCatccatc 15268
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-hom 14568  df-cco 14569  df-homf 14914  df-comf 14915  df-catc 15269
This theorem is referenced by: (None)
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