MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resscatc Structured version   Unicode version

Theorem resscatc 15501
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 2382 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
4 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
53, 4ssexd 4512 . . . . . . 7  |-  ( ph  ->  V  e.  _V )
65adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  V  e.  _V )
7 eqid 2382 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
8 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( V  i^i  Cat ) )
91, 2, 5catcbas 15493 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  ( V  i^i  Cat ) )
109adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( Base `  D )  =  ( V  i^i  Cat ) )
118, 10eleqtrrd 2473 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  D
) )
12 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( V  i^i  Cat ) )
1312, 10eleqtrrd 2473 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
141, 2, 6, 7, 11, 13catchom 15495 . . . . 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 2382 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
173adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  U  e.  W )
18 eqid 2382 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
1915, 16, 3catcbas 15493 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2019ineq2d 3614 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
21 inass 3622 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2220, 21syl6reqr 2442 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
23 df-ss 3403 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
244, 23sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2524ineq1d 3613 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
26 eqid 2382 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2726, 16ressbas 14691 . . . . . . . . . . 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 2431 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3026, 16ressbasss 14693 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3129, 30syl6eqss 3467 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  Cat )  C_  ( Base `  C
) )
3231adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( V  i^i  Cat )  C_  ( Base `  C )
)
3332, 8sseldd 3418 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3432, 12sseldd 3418 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3515, 16, 17, 18, 33, 34catchom 15495 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  C
) y )  =  ( x  Func  y
) )
3626, 18resshom 14825 . . . . . . 7  |-  ( V  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  ( Cs  V ) ) )
375, 36syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  ( Cs  V ) ) )
3837oveqdr 6220 . . . . 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 2430 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x ( Hom  `  ( Cs  V ) ) y )  =  ( x ( Hom  `  D
) y ) )
4039ralrimivva 2803 . . 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 2382 . . . 4  |-  ( Hom  `  ( Cs  V ) )  =  ( Hom  `  ( Cs  V ) )
429eqcomd 2390 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4341, 7, 29, 42homfeq 15100 . . 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 723 . . . . . . . 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 2382 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
47 simplr1 1036 . . . . . . . . 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 723 . . . . . . . . 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 2473 . . . . . . . 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 1037 . . . . . . . . 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 2473 . . . . . . . 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 1038 . . . . . . . . 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 2473 . . . . . . . 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 754 . . . . . . . . 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 15495 . . . . . . . . 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 2472 . . . . . . . 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 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 ) ) )  ->  g  e.  ( y ( Hom  `  D
) z ) )
581, 2, 45, 7, 51, 53catchom 15495 . . . . . . . . 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 2472 . . . . . . . 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 15497 . . . . . . 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 723 . . . . . . . 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 2382 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
6331ad2antrr 723 . . . . . . . . 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 3418 . . . . . . . 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 3418 . . . . . . . 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 3418 . . . . . . . 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 15497 . . . . . . 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 14826 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  (comp `  C )  =  (comp `  ( Cs  V ) ) )
695, 68syl 16 . . . . . . . . . 10  |-  ( ph  ->  (comp `  C )  =  (comp `  ( Cs  V
) ) )
7069ad2antrr 723 . . . . . . . . 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 6213 . . . . . . . 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 6213 . . . . . . 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 2429 . . . . . 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 2803 . . . . 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 2804 . . . 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 2382 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7744eqcomd 2390 . . . . 5  |-  ( ph  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  V ) ) )
7846, 76, 7, 42, 29, 77comfeq 15112 . . . 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 2390 . 2  |-  ( ph  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
8144, 80jca 530 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    i^i cin 3388    C_ wss 3389   <.cop 3950   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   Hom chom 14713  compcco 14714   Catccat 15071   Hom f chomf 15073  compfccomf 15074    Func cfunc 15260    o.func ccofu 15262  CatCatccatc 15490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-hom 14726  df-cco 14727  df-homf 15077  df-comf 15078  df-catc 15491
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator