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Theorem resscatc 14965
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 2437 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
4 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
53, 4ssexd 4432 . . . . . . 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 2437 . . . . . 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 14957 . . . . . . . 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 2514 . . . . . 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 2514 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
141, 2, 6, 7, 11, 13catchom 14959 . . . . 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 2437 . . . . . 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 2437 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
1915, 16, 3catcbas 14957 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2019ineq2d 3545 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
21 inass 3553 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2220, 21syl6reqr 2488 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
23 df-ss 3335 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
244, 23sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2524ineq1d 3544 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
26 eqid 2437 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2726, 16ressbas 14220 . . . . . . . . . . 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 2477 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3026, 16ressbasss 14222 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3129, 30syl6eqss 3399 . . . . . . . 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 3350 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3432, 12sseldd 3350 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3515, 16, 17, 18, 33, 34catchom 14959 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x  Func  y
) )
3626, 18resshom 14349 . . . . . . 7  |-  ( V  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  ( Cs  V ) ) )
375, 36syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  ( Cs  V ) ) )
3837proplem3 14621 . . . . 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 2476 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) )
4039ralrimivva 2802 . . 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 2437 . . . 4  |-  ( Hom  `  ( Cs  V ) )  =  ( Hom  `  ( Cs  V ) )
429eqcomd 2442 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4341, 7, 29, 42homfeq 14625 . . 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 2437 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
47 simplr1 1030 . . . . . . . . 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 2514 . . . . . . . 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 1031 . . . . . . . . 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 2514 . . . . . . . 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 1032 . . . . . . . . 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 2514 . . . . . . . 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 14959 . . . . . . . . 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 2513 . . . . . . . 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 14959 . . . . . . . . 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 2513 . . . . . . . 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 14961 . . . . . . 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 2437 . . . . . . . 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 3350 . . . . . . . 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 3350 . . . . . . . 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 3350 . . . . . . . 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 14961 . . . . . . 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 14350 . . . . . . . . . . 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 6103 . . . . . . . 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 6103 . . . . . . 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 2475 . . . . . 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 2802 . . . . 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 2803 . . . 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 2437 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7744eqcomd 2442 . . . . 5  |-  ( ph  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  V ) ) )
7846, 76, 7, 42, 29, 77comfeq 14637 . . . 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 2442 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2709   _Vcvv 2966    i^i cin 3320    C_ wss 3321   <.cop 3876   ` cfv 5411  (class class class)co 6086   Basecbs 14166   ↾s cress 14167   Hom chom 14241  compcco 14242   Catccat 14594   Hom f chomf 14596  compfccomf 14597    Func cfunc 14756    o.func ccofu 14758  CatCatccatc 14954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-hom 14254  df-cco 14255  df-homf 14600  df-comf 14601  df-catc 14955
This theorem is referenced by: (None)
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