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Theorem rescabs 15816
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 2471 . . . 4  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 6336 . . . . 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 4543 . . . 4  |-  ( ph  ->  T  e.  _V )
7 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
81, 3, 6, 7rescval2 15811 . . 3  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
9 simpr 468 . . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
12 eqid 2471 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )
13 baseid 15247 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
14 1re 9660 . . . . . . . . . . 11  |-  1  e.  RR
15 1nn 10642 . . . . . . . . . . . 12  |-  1  e.  NN
16 4nn0 10912 . . . . . . . . . . . 12  |-  4  e.  NN0
17 1nn0 10909 . . . . . . . . . . . 12  |-  1  e.  NN0
18 1lt10 10843 . . . . . . . . . . . 12  |-  1  <  10
1915, 16, 17, 18declti 11099 . . . . . . . . . . 11  |-  1  < ; 1
4
2014, 19ltneii 9765 . . . . . . . . . 10  |-  1  =/= ; 1 4
21 basendx 15251 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
22 homndx 15390 . . . . . . . . . . 11  |-  ( Hom  `  ndx )  = ; 1 4
2321, 22neeq12i 2709 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2420, 23mpbir 214 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  ( Hom  `  ndx )
2513, 24setsnid 15243 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2612, 25ressid2 15255 . . . . . . 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 1292 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
2827oveq1d 6323 . . . . 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 6336 . . . . . 6  |-  ( Cs  S )  e.  _V
30 xpexg 6612 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  T  e.  _V )  ->  ( T  X.  T
)  e.  _V )
316, 6, 30syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( T  X.  T
)  e.  _V )
32 fnex 6148 . . . . . . . 8  |-  ( ( J  Fn  ( T  X.  T )  /\  ( T  X.  T
)  e.  _V )  ->  J  e.  _V )
337, 31, 32syl2anc 673 . . . . . . 7  |-  ( ph  ->  J  e.  _V )
3433adantr 472 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
35 setsabs 15230 . . . . . 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 676 . . . . 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 2471 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
38 eqid 2471 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
3937, 38ressbas 15257 . . . . . . . . . . . . 13  |-  ( S  e.  W  ->  ( S  i^i  ( Base `  C
) )  =  (
Base `  ( Cs  S
) ) )
404, 39syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  i^i  ( Base `  C ) )  =  ( Base `  ( Cs  S ) ) )
4140sseq1d 3445 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  i^i  ( Base `  C )
)  C_  T  <->  ( Base `  ( Cs  S ) )  C_  T ) )
4241biimpar 493 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  T
)
43 inss2 3644 . . . . . . . . . . 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 3647 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( T  i^i  ( Base `  C
) ) )
465adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
47 ssrin 3648 . . . . . . . . . 10  |-  ( T 
C_  S  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
4846, 47syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
4945, 48eqssd 3435 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5049oveq2d 6324 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
514adantr 472 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
5238ressinbas 15263 . . . . . . . 8  |-  ( S  e.  W  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5351, 52syl 17 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5438ressinbas 15263 . . . . . . . 8  |-  ( T  e.  _V  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
5511, 54syl 17 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
5650, 53, 553eqtr4d 2515 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
5756oveq1d 6323 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
5828, 36, 573eqtrd 2509 . . . 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 468 . . . . . . . 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 472 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
6212, 25ressval2 15256 . . . . . . . 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 1292 . . . . . . 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 2697 . . . . . . . . 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 6612 . . . . . . . . . . 11  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
694, 4, 68syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
70 fnex 6148 . . . . . . . . . 10  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
7167, 69, 70syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
7271adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  H  e.  _V )
73 fvex 5889 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7473inex2 4538 . . . . . . . . 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 5889 . . . . . . . . 9  |-  ( Hom  `  ndx )  e.  _V
77 fvex 5889 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
7876, 77setscom 15231 . . . . . . . 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 1293 . . . . . . 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 2471 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
81 eqid 2471 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8280, 81ressval2 15256 . . . . . . . . . 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 1292 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
844adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
855adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
86 ressabs 15266 . . . . . . . . . 10  |-  ( ( S  e.  W  /\  T  C_  S )  -> 
( ( Cs  S )s  T )  =  ( Cs  T ) )
8784, 85, 86syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( Cs  T ) )
8883, 87eqtr3d 2507 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
8988oveq1d 6323 . . . . . . 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 2509 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
9190oveq1d 6323 . . . . 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 6336 . . . . . 6  |-  ( Cs  T )  e.  _V
9333adantr 472 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
94 setsabs 15230 . . . . . 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 676 . . . . 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 2505 . . . 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 808 . . 3  |-  ( ph  ->  ( ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
988, 97eqtrd 2505 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
99 eqid 2471 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
100 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
10199, 100, 4, 67rescval2 15811 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
102101oveq1d 6323 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
103 eqid 2471 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
104103, 100, 6, 7rescval2 15811 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
10598, 102, 1043eqtr4d 2515 1  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    i^i cin 3389    C_ wss 3390   <.cop 3965    X. cxp 4837    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   1c1 9558   4c4 10683  ;cdc 11074   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾s cress 15200   Hom chom 15279    |`cat cresc 15791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-hom 15292  df-resc 15794
This theorem is referenced by:  subsubc  15836  fldc  40593  fldcALTV  40612
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