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Theorem isummolem2 10835
Description: Lemma for isummo 10836. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 8-Sep-2022.)
Hypotheses
Ref Expression
isummo.1 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
isummo.2 |- ( ( ph /\ k e. A ) -> B e. CC )
isummolem2.g |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )
Assertion
Ref Expression
isummolem2 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F , CC ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
Distinct variable groups:   k, n, A   n, F   ph, k, n   A, f, j, m, k, n   B, n   f, F, k, m   ph, f, m   x, f, k, m, n   y, f, m
Allowed substitution hints:   ph( x, y, j)   A( x, y)   B( x, y, f, j, k, m)   F( x, y, j)   G( x, y, f, j, k, m, n)
Proof of Theorem isummolem2
Dummy variables a g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5320 . . . . 5 |- ( m = a -> ( ZZ>= ` m ) = ( ZZ>= ` a ) )
21sseq2d 3057 . . . 4 |- ( m = a -> ( A C_ ( ZZ>= ` m ) <-> A C_ ( ZZ>= ` a ) ) )
31raleqdv 2571 . . . 4 |- ( m = a -> ( A. j e. ( ZZ>= ` m )DECID j e. A <-> A. j e. ( ZZ>= ` a )DECID j e. A ) )
4 iseqeq1 9921 . . . . 5 |- ( m = a -> seq m ( + , F , CC ) = seq a ( + , F , CC ) )
54breq1d 3863 . . . 4 |- ( m = a -> ( seq m ( + , F , CC ) ~~> x <-> seq a ( + , F , CC ) ~~> x ) )
62, 3, 53anbi123d 1249 . . 3 |- ( m = a -> ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F , CC ) ~~> x ) <-> ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) )
76cbvrexv 2594 . 2 |- ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F , CC ) ~~> x ) <-> E. a e. ZZ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) )
8 simplr3 988 . . . . . . . . 9 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq a ( + , F , CC ) ~~> x )
9 simplr1 986 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ( ZZ>= ` a ) )
10 uzssz 9101 . . . . . . . . . . . 12 |- ( ZZ>= ` a ) C_ ZZ
119, 10syl6ss 3040 . . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A C_ ZZ )
12 1zzd 8840 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> 1 e. ZZ )
13 simprl 499 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> m e. NN )
1413nnzd 8930 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> m e. ZZ )
1512, 14fzfigd 9901 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) e. Fin )
16 simprr 500 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
17 f1oeng 6530 . . . . . . . . . . . . . 14 |- ( ( ( 1 ... m ) e. Fin /\ f : ( 1 ... m ) -1-1-onto-> A ) -> ( 1 ... m ) ~~ A )
1815, 16, 17syl2anc 404 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( 1 ... m ) ~~ A )
1918ensymd 6556 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A ~~ ( 1 ... m ) )
20 enfii 6646 . . . . . . . . . . . 12 |- ( ( ( 1 ... m ) e. Fin /\ A ~~ ( 1 ... m ) ) -> A e. Fin )
2115, 19, 20syl2anc 404 . . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> A e. Fin )
22 zfz1iso 10309 . . . . . . . . . . 11 |- ( ( A C_ ZZ /\ A e. Fin ) -> E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
2311, 21, 22syl2anc 404 . . . . . . . . . 10 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
24 isummo.1 . . . . . . . . . . . . 13 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
25 simplll 501 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> ph )
26 isummo.2 . . . . . . . . . . . . . 14 |- ( ( ph /\ k e. A ) -> B e. CC )
2725, 26sylan 278 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. A ) -> B e. CC )
28 eleq1w 2149 . . . . . . . . . . . . . . 15 |- ( j = k -> ( j e. A <-> k e. A ) )
2928dcbid 787 . . . . . . . . . . . . . 14 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
30 simpr2 951 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) -> A. j e. ( ZZ>= ` a )DECID j e. A )
3130ad2antrr 473 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> A. j e. ( ZZ>= ` a )DECID j e. A )
32 simpr 109 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> k e. ( ZZ>= ` a ) )
3329, 31, 32rspcdva 2730 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) /\ k e. ( ZZ>= ` a ) ) -> DECID k e. A )
34 isummolem2.g . . . . . . . . . . . . 13 |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )
35 eqid 2089 . . . . . . . . . . . . 13 |- ( n e. NN |-> if ( n <_ m , [_ ( g ` n ) / k ]_ B , 0 ) ) = ( n e. NN |-> if ( n <_ m , [_ ( g ` n ) / k ]_ B , 0 ) )
36 simprll 505 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> m e. NN )
37 simpllr 502 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> a e. ZZ )
38 simplr1 986 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> A C_ ( ZZ>= ` a ) )
39 simprlr 506 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> f : ( 1 ... m ) -1-1-onto-> A )
40 simprr 500 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
4124, 27, 33, 34, 35, 36, 37, 38, 39, 40isummolem2a 10834 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) /\ g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) ) ) -> seq a ( + , F , CC ) ~~> ( seq 1 ( + , G , CC ) ` m ) )
4241expr 368 . . . . . . . . . . 11 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) -> seq a ( + , F , CC ) ~~> ( seq 1 ( + , G , CC ) ` m ) ) )
4342exlimdv 1748 . . . . . . . . . 10 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> ( E. g g Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) -> seq a ( + , F , CC ) ~~> ( seq 1 ( + , G , CC ) ` m ) ) )
4423, 43mpd 13 . . . . . . . . 9 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> seq a ( + , F , CC ) ~~> ( seq 1 ( + , G , CC ) ` m ) )
45 climuni 10744 . . . . . . . . 9 |- ( ( seq a ( + , F , CC ) ~~> x /\ seq a ( + , F , CC ) ~~> ( seq 1 ( + , G , CC ) ` m ) ) -> x = ( seq 1 ( + , G , CC ) ` m ) )
468, 44, 45syl2anc 404 . . . . . . . 8 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ ( m e. NN /\ f : ( 1 ... m ) -1-1-onto-> A ) ) -> x = ( seq 1 ( + , G , CC ) ` m ) )
4746anassrs 393 . . . . . . 7 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> x = ( seq 1 ( + , G , CC ) ` m ) )
48 eqeq2 2098 . . . . . . 7 |- ( y = ( seq 1 ( + , G , CC ) ` m ) -> ( x = y <-> x = ( seq 1 ( + , G , CC ) ` m ) ) )
4947, 48syl5ibrcom 156 . . . . . 6 |- ( ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ m e. NN ) /\ f : ( 1 ... m ) -1-1-onto-> A ) -> ( y = ( seq 1 ( + , G , CC ) ` m ) -> x = y ) )
5049expimpd 356 . . . . 5 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ m e. NN ) -> ( ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
5150exlimdv 1748 . . . 4 |- ( ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) /\ m e. NN ) -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
5251rexlimdva 2491 . . 3 |- ( ( ( ph /\ a e. ZZ ) /\ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
5352r19.29an 2513 . 2 |- ( ( ph /\ E. a e. ZZ ( A C_ ( ZZ>= ` a ) /\ A. j e. ( ZZ>= ` a )DECID j e. A /\ seq a ( + , F , CC ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
547, 53sylan2b 282 1 |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F , CC ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G , CC ) ` m ) ) -> x = y ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103  DECID wdc 781   /\ w3a 925   = wceq 1290   E.wex 1427   e. wcel 1439   A.wral 2360   E.wrex 2361   [_csb 2936   C_ wss 3002   ifcif 3399   class class class wbr 3853   |-> cmpt 3907   -1-1-onto->wf1o 5029   ` cfv 5030   Isom wiso 5031  (class class class)co 5668   ~~ cen 6511   Fincfn 6513   CCcc 7411   0cc0 7413   1c1 7414   + caddc 7416   < clt 7585   <_ cle 7586   NNcn 8485   ZZcz 8813   ZZ>=cuz 9082   ...cfz 9487   seqcseq4 9914  ♯chash 10246   ~~> cli 10729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730
This theorem is referenced by:  isummo  10836
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