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Theorem rescabs 13988
Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
rescabs.c  |-  ( ph  ->  C  e.  V )
rescabs.h  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
rescabs.j  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
rescabs.s  |-  ( ph  ->  S  e.  W )
rescabs.t  |-  ( ph  ->  T  C_  S )
Assertion
Ref Expression
rescabs  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )

Proof of Theorem rescabs
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 6065 . . . . 5  |-  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
32a1i 11 . . . 4  |-  ( ph  ->  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
4 rescabs.s . . . . 5  |-  ( ph  ->  S  e.  W )
5 rescabs.t . . . . 5  |-  ( ph  ->  T  C_  S )
64, 5ssexd 4310 . . . 4  |-  ( ph  ->  T  e.  _V )
7 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
81, 3, 6, 7rescval2 13983 . . 3  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
9 simpr 448 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Base `  ( Cs  S ) )  C_  T )
102a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
116adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
12 eqid 2404 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )
13 baseid 13466 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
14 1re 9046 . . . . . . . . . . 11  |-  1  e.  RR
15 1nn 9967 . . . . . . . . . . . 12  |-  1  e.  NN
16 4nn0 10196 . . . . . . . . . . . 12  |-  4  e.  NN0
17 1nn0 10193 . . . . . . . . . . . 12  |-  1  e.  NN0
18 1lt10 10142 . . . . . . . . . . . 12  |-  1  <  10
1915, 16, 17, 18declti 10363 . . . . . . . . . . 11  |-  1  < ; 1
4
2014, 19ltneii 9142 . . . . . . . . . 10  |-  1  =/= ; 1 4
21 basendx 13469 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
22 homndx 13597 . . . . . . . . . . 11  |-  (  Hom  `  ndx )  = ; 1 4
2321, 22neeq12i 2579 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2420, 23mpbir 201 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
2513, 24setsnid 13464 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
2612, 25ressid2 13472 . . . . . . 7  |-  ( ( ( Base `  ( Cs  S ) )  C_  T  /\  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V  /\  T  e.  _V )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
279, 10, 11, 26syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
2827oveq1d 6055 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
29 ovex 6065 . . . . . 6  |-  ( Cs  S )  e.  _V
30 xpexg 4948 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  T  e.  _V )  ->  ( T  X.  T
)  e.  _V )
316, 6, 30syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( T  X.  T
)  e.  _V )
32 fnex 5920 . . . . . . . 8  |-  ( ( J  Fn  ( T  X.  T )  /\  ( T  X.  T
)  e.  _V )  ->  J  e.  _V )
337, 31, 32syl2anc 643 . . . . . . 7  |-  ( ph  ->  J  e.  _V )
3433adantr 452 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
35 setsabs 13451 . . . . . 6  |-  ( ( ( Cs  S )  e.  _V  /\  J  e.  _V )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
3629, 34, 35sylancr 645 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
37 eqid 2404 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
38 eqid 2404 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
3937, 38ressbas 13474 . . . . . . . . . . . . 13  |-  ( S  e.  W  ->  ( S  i^i  ( Base `  C
) )  =  (
Base `  ( Cs  S
) ) )
404, 39syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  i^i  ( Base `  C ) )  =  ( Base `  ( Cs  S ) ) )
4140sseq1d 3335 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  i^i  ( Base `  C )
)  C_  T  <->  ( Base `  ( Cs  S ) )  C_  T ) )
4241biimpar 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  T
)
43 inss2 3522 . . . . . . . . . . 11  |-  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C )
4443a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C ) )
4542, 44ssind 3525 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( T  i^i  ( Base `  C
) ) )
465adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
47 ssrin 3526 . . . . . . . . . 10  |-  ( T 
C_  S  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
4846, 47syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
4945, 48eqssd 3325 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5049oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
514adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
5238ressinbas 13480 . . . . . . . 8  |-  ( S  e.  W  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5351, 52syl 16 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5438ressinbas 13480 . . . . . . . 8  |-  ( T  e.  _V  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
5511, 54syl 16 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
5650, 53, 553eqtr4d 2446 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
5756oveq1d 6055 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
5828, 36, 573eqtrd 2440 . . . 4  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
59 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  -.  ( Base `  ( Cs  S ) )  C_  T )
602a1i 11 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
616adantr 452 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
6212, 25ressval2 13473 . . . . . . . 8  |-  ( ( -.  ( Base `  ( Cs  S ) )  C_  T  /\  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V  /\  T  e.  _V )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
6359, 60, 61, 62syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
6429a1i 11 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  e.  _V )
6524necomi 2649 . . . . . . . . 9  |-  (  Hom  `  ndx )  =/=  ( Base `  ndx )
6665a1i 11 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  (  Hom  `  ndx )  =/=  ( Base `  ndx ) )
67 rescabs.h . . . . . . . . . 10  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
68 xpexg 4948 . . . . . . . . . . 11  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
694, 4, 68syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
70 fnex 5920 . . . . . . . . . 10  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
7167, 69, 70syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
7271adantr 452 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  H  e.  _V )
73 fvex 5701 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7473inex2 4305 . . . . . . . . 9  |-  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V
7574a1i 11 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V )
76 fvex 5701 . . . . . . . . 9  |-  (  Hom  `  ndx )  e.  _V
77 fvex 5701 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
7876, 77setscom 13452 . . . . . . . 8  |-  ( ( ( ( Cs  S )  e.  _V  /\  (  Hom  `  ndx )  =/=  ( Base `  ndx ) )  /\  ( H  e.  _V  /\  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V ) )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )  =  ( ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. ) sSet  <.
(  Hom  `  ndx ) ,  H >. ) )
7964, 66, 72, 75, 78syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  (
( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
80 eqid 2404 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
81 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8280, 81ressval2 13473 . . . . . . . . . 10  |-  ( ( -.  ( Base `  ( Cs  S ) )  C_  T  /\  ( Cs  S )  e.  _V  /\  T  e.  _V )  ->  (
( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
8359, 64, 61, 82syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
844adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
855adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
86 ressabs 13482 . . . . . . . . . 10  |-  ( ( S  e.  W  /\  T  C_  S )  -> 
( ( Cs  S )s  T )  =  ( Cs  T ) )
8784, 85, 86syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( Cs  T ) )
8883, 87eqtr3d 2438 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
8988oveq1d 6055 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. ) sSet  <. (  Hom  `  ndx ) ,  H >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9063, 79, 893eqtrd 2440 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9190oveq1d 6055 . . . . 5  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
92 ovex 6065 . . . . . 6  |-  ( Cs  T )  e.  _V
9333adantr 452 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
94 setsabs 13451 . . . . . 6  |-  ( ( ( Cs  T )  e.  _V  /\  J  e.  _V )  ->  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
9592, 93, 94sylancr 645 . . . . 5  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
9691, 95eqtrd 2436 . . . 4  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
9758, 96pm2.61dan 767 . . 3  |-  ( ph  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
988, 97eqtrd 2436 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
99 eqid 2404 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
100 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
10199, 100, 4, 67rescval2 13983 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
102101oveq1d 6055 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
103 eqid 2404 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
104103, 100, 6, 7rescval2 13983 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
10598, 102, 1043eqtr4d 2446 1  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    i^i cin 3279    C_ wss 3280   <.cop 3777    X. cxp 4835    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1c1 8947   4c4 10007  ;cdc 10338   ndxcnx 13421   sSet csts 13422   Basecbs 13424   ↾s cress 13425    Hom chom 13495    |`cat cresc 13963
This theorem is referenced by:  subsubc  14005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-hom 13508  df-resc 13966
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