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Theorem rescabs 15063
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 2467 . . . 4  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 6309 . . . . 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 4594 . . . 4  |-  ( ph  ->  T  e.  _V )
7 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
81, 3, 6, 7rescval2 15058 . . 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 2467 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )
13 baseid 14536 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
14 1re 9595 . . . . . . . . . . 11  |-  1  e.  RR
15 1nn 10547 . . . . . . . . . . . 12  |-  1  e.  NN
16 4nn0 10814 . . . . . . . . . . . 12  |-  4  e.  NN0
17 1nn0 10811 . . . . . . . . . . . 12  |-  1  e.  NN0
18 1lt10 10746 . . . . . . . . . . . 12  |-  1  <  10
1915, 16, 17, 18declti 11001 . . . . . . . . . . 11  |-  1  < ; 1
4
2014, 19ltneii 9697 . . . . . . . . . 10  |-  1  =/= ; 1 4
21 basendx 14540 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
22 homndx 14670 . . . . . . . . . . 11  |-  ( Hom  `  ndx )  = ; 1 4
2321, 22neeq12i 2756 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2420, 23mpbir 209 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
2513, 24setsnid 14532 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2612, 25ressid2 14543 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2827oveq1d 6299 . . . . 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 6309 . . . . . 6  |-  ( Cs  S )  e.  _V
30 xpexg 6586 . . . . . . . . 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 6127 . . . . . . . 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 14519 . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
38 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
3937, 38ressbas 14545 . . . . . . . . . . . . 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 3531 . . . . . . . . . . 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 3719 . . . . . . . . . . 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 3722 . . . . . . . . 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 3723 . . . . . . . . . 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 3521 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5049oveq2d 6300 . . . . . . 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 14551 . . . . . . . 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 14551 . . . . . . . 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 2518 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
5756oveq1d 6299 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
5828, 36, 573eqtrd 2512 . . . 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 14544 . . . . . . . 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 1228 . . . . . . 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 2737 . . . . . . . . 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 6586 . . . . . . . . . . 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 6127 . . . . . . . . . 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 5876 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7473inex2 4589 . . . . . . . . 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 5876 . . . . . . . . 9  |-  ( Hom  `  ndx )  e.  _V
77 fvex 5876 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
7876, 77setscom 14520 . . . . . . . 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 1229 . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
81 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8280, 81ressval2 14544 . . . . . . . . . 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 1228 . . . . . . . . 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 14553 . . . . . . . . . 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 2510 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
8988oveq1d 6299 . . . . . . 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 2512 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
9190oveq1d 6299 . . . . 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 6309 . . . . . 6  |-  ( Cs  T )  e.  _V
9333adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
94 setsabs 14519 . . . . . 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 2508 . . . 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 2508 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
99 eqid 2467 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
100 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
10199, 100, 4, 67rescval2 15058 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
102101oveq1d 6299 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
103 eqid 2467 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
104103, 100, 6, 7rescval2 15058 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
10598, 102, 1043eqtr4d 2518 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    i^i cin 3475    C_ wss 3476   <.cop 4033    X. cxp 4997    Fn wfn 5583   ` cfv 5588  (class class class)co 6284   1c1 9493   4c4 10587  ;cdc 10976   ndxcnx 14487   sSet csts 14488   Basecbs 14490   ↾s cress 14491   Hom chom 14566    |`cat cresc 15038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-hom 14579  df-resc 15041
This theorem is referenced by:  subsubc  15080
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