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Theorem rescabs 13953
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 2380 . . . 4  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 6038 . . . . 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 4284 . . . 4  |-  ( ph  ->  T  e.  _V )
7 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
81, 3, 6, 7rescval2 13948 . . 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 2380 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )
13 baseid 13431 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
14 1re 9016 . . . . . . . . . . 11  |-  1  e.  RR
15 1nn 9936 . . . . . . . . . . . 12  |-  1  e.  NN
16 4nn0 10165 . . . . . . . . . . . 12  |-  4  e.  NN0
17 1nn0 10162 . . . . . . . . . . . 12  |-  1  e.  NN0
18 1lt10 10111 . . . . . . . . . . . 12  |-  1  <  10
1915, 16, 17, 18declti 10332 . . . . . . . . . . 11  |-  1  < ; 1
4
2014, 19ltneii 9110 . . . . . . . . . 10  |-  1  =/= ; 1 4
21 basendx 13434 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
22 homndx 13562 . . . . . . . . . . 11  |-  (  Hom  `  ndx )  = ; 1 4
2321, 22neeq12i 2555 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2420, 23mpbir 201 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
2513, 24setsnid 13429 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
2612, 25ressid2 13437 . . . . . . 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 6028 . . . . 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 6038 . . . . . 6  |-  ( Cs  S )  e.  _V
30 xpexg 4922 . . . . . . . . 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 5893 . . . . . . . 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 13416 . . . . . 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 2380 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
38 eqid 2380 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
3937, 38ressbas 13439 . . . . . . . . . . . . 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 3311 . . . . . . . . . . 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 3498 . . . . . . . . . . 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 3501 . . . . . . . . 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 3502 . . . . . . . . . 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 3301 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5049oveq2d 6029 . . . . . . 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 13445 . . . . . . . 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 13445 . . . . . . . 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 2422 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
5756oveq1d 6028 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
5828, 36, 573eqtrd 2416 . . . 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 13438 . . . . . . . 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 2625 . . . . . . . . 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 4922 . . . . . . . . . . 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 5893 . . . . . . . . . 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 5675 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7473inex2 4279 . . . . . . . . 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 5675 . . . . . . . . 9  |-  (  Hom  `  ndx )  e.  _V
77 fvex 5675 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
7876, 77setscom 13417 . . . . . . . 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 2380 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
81 eqid 2380 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8280, 81ressval2 13438 . . . . . . . . . 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 13447 . . . . . . . . . 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 2414 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
8988oveq1d 6028 . . . . . . 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 2416 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9190oveq1d 6028 . . . . 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 6038 . . . . . 6  |-  ( Cs  T )  e.  _V
9333adantr 452 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
94 setsabs 13416 . . . . . 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 2412 . . . 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 2412 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
99 eqid 2380 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
100 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
10199, 100, 4, 67rescval2 13948 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
102101oveq1d 6028 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
103 eqid 2380 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
104103, 100, 6, 7rescval2 13948 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
10598, 102, 1043eqtr4d 2422 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 1717    =/= wne 2543   _Vcvv 2892    i^i cin 3255    C_ wss 3256   <.cop 3753    X. cxp 4809    Fn wfn 5382   ` cfv 5387  (class class class)co 6013   1c1 8917   4c4 9976  ;cdc 10307   ndxcnx 13386   sSet csts 13387   Basecbs 13389   ↾s cress 13390    Hom chom 13460    |`cat cresc 13928
This theorem is referenced by:  subsubc  13970
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-hom 13473  df-resc 13931
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