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Theorem rescabs 14738
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 2438 . . . 4  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 6111 . . . . 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 4434 . . . 4  |-  ( ph  ->  T  e.  _V )
7 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
81, 3, 6, 7rescval2 14733 . . 3  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
9 simpr 461 . . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
12 eqid 2438 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )
13 baseid 14212 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
14 1re 9377 . . . . . . . . . . 11  |-  1  e.  RR
15 1nn 10325 . . . . . . . . . . . 12  |-  1  e.  NN
16 4nn0 10590 . . . . . . . . . . . 12  |-  4  e.  NN0
17 1nn0 10587 . . . . . . . . . . . 12  |-  1  e.  NN0
18 1lt10 10524 . . . . . . . . . . . 12  |-  1  <  10
1915, 16, 17, 18declti 10772 . . . . . . . . . . 11  |-  1  < ; 1
4
2014, 19ltneii 9479 . . . . . . . . . 10  |-  1  =/= ; 1 4
21 basendx 14215 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
22 homndx 14345 . . . . . . . . . . 11  |-  ( Hom  `  ndx )  = ; 1 4
2321, 22neeq12i 2615 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2420, 23mpbir 209 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
2513, 24setsnid 14208 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2612, 25ressid2 14218 . . . . . . 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 1218 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2827oveq1d 6101 . . . . 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 6111 . . . . . 6  |-  ( Cs  S )  e.  _V
30 xpexg 6502 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  T  e.  _V )  ->  ( T  X.  T
)  e.  _V )
316, 6, 30syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( T  X.  T
)  e.  _V )
32 fnex 5939 . . . . . . . 8  |-  ( ( J  Fn  ( T  X.  T )  /\  ( T  X.  T
)  e.  _V )  ->  J  e.  _V )
337, 31, 32syl2anc 661 . . . . . . 7  |-  ( ph  ->  J  e.  _V )
3433adantr 465 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
35 setsabs 14195 . . . . . 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 663 . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
38 eqid 2438 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
3937, 38ressbas 14220 . . . . . . . . . . . . 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 3378 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  i^i  ( Base `  C )
)  C_  T  <->  ( Base `  ( Cs  S ) )  C_  T ) )
4241biimpar 485 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  T
)
43 inss2 3566 . . . . . . . . . . 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 3569 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( T  i^i  ( Base `  C
) ) )
465adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
47 ssrin 3570 . . . . . . . . . 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 3368 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5049oveq2d 6102 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
514adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
5238ressinbas 14226 . . . . . . . 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 14226 . . . . . . . 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 2480 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
5756oveq1d 6101 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
5828, 36, 573eqtrd 2474 . . . 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 461 . . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
6212, 25ressval2 14219 . . . . . . . 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 1218 . . . . . . 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 2689 . . . . . . . . 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 6502 . . . . . . . . . . 11  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
694, 4, 68syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
70 fnex 5939 . . . . . . . . . 10  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
7167, 69, 70syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
7271adantr 465 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  H  e.  _V )
73 fvex 5696 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7473inex2 4429 . . . . . . . . 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 5696 . . . . . . . . 9  |-  ( Hom  `  ndx )  e.  _V
77 fvex 5696 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
7876, 77setscom 14196 . . . . . . . 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 1219 . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
81 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8280, 81ressval2 14219 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
844adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
855adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
86 ressabs 14228 . . . . . . . . . 10  |-  ( ( S  e.  W  /\  T  C_  S )  -> 
( ( Cs  S )s  T )  =  ( Cs  T ) )
8784, 85, 86syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( Cs  T ) )
8883, 87eqtr3d 2472 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
8988oveq1d 6101 . . . . . . 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 2474 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
9190oveq1d 6101 . . . . 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 6111 . . . . . 6  |-  ( Cs  T )  e.  _V
9333adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
94 setsabs 14195 . . . . . 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 663 . . . . 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 2470 . . . 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 789 . . 3  |-  ( ph  ->  ( ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
988, 97eqtrd 2470 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
99 eqid 2438 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
100 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
10199, 100, 4, 67rescval2 14733 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
102101oveq1d 6101 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
103 eqid 2438 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
104103, 100, 6, 7rescval2 14733 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
10598, 102, 1043eqtr4d 2480 1  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967    i^i cin 3322    C_ wss 3323   <.cop 3878    X. cxp 4833    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   1c1 9275   4c4 10365  ;cdc 10747   ndxcnx 14163   sSet csts 14164   Basecbs 14166   ↾s cress 14167   Hom chom 14241    |`cat cresc 14713
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  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-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-hom 14254  df-resc 14716
This theorem is referenced by:  subsubc  14755
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