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Mirrors > Home > ILE Home > Th. List > iseqval | Unicode version |
Description: Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
iseqval.1 | frec |
iseqval.f | |
iseqval.pl |
Ref | Expression |
---|---|
iseqval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqval.1 | . . . 4 frec | |
2 | simprl 483 | . . . . . . . . 9 | |
3 | simprr 484 | . . . . . . . . 9 | |
4 | iseqval.pl | . . . . . . . . . . . 12 | |
5 | 4 | caovclg 5653 | . . . . . . . . . . 11 |
6 | 5 | adantlr 446 | . . . . . . . . . 10 |
7 | iseqval.f | . . . . . . . . . . . . . 14 | |
8 | 7 | ralrimiva 2392 | . . . . . . . . . . . . 13 |
9 | fveq2 5178 | . . . . . . . . . . . . . . 15 | |
10 | 9 | eleq1d 2106 | . . . . . . . . . . . . . 14 |
11 | 10 | cbvralv 2533 | . . . . . . . . . . . . 13 |
12 | 8, 11 | sylib 127 | . . . . . . . . . . . 12 |
13 | 12 | adantr 261 | . . . . . . . . . . 11 |
14 | peano2uz 8526 | . . . . . . . . . . . . 13 | |
15 | fveq2 5178 | . . . . . . . . . . . . . . 15 | |
16 | 15 | eleq1d 2106 | . . . . . . . . . . . . . 14 |
17 | 16 | rspcv 2652 | . . . . . . . . . . . . 13 |
18 | 14, 17 | syl 14 | . . . . . . . . . . . 12 |
19 | 18 | ad2antrl 459 | . . . . . . . . . . 11 |
20 | 13, 19 | mpd 13 | . . . . . . . . . 10 |
21 | 6, 3, 20 | caovcld 5654 | . . . . . . . . 9 |
22 | oveq1 5519 | . . . . . . . . . . . 12 | |
23 | 22 | fveq2d 5182 | . . . . . . . . . . 11 |
24 | 23 | oveq2d 5528 | . . . . . . . . . 10 |
25 | oveq1 5519 | . . . . . . . . . 10 | |
26 | eqid 2040 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | ovmpt2g 5635 | . . . . . . . . 9 |
28 | 2, 3, 21, 27 | syl3anc 1135 | . . . . . . . 8 |
29 | 28 | 3impb 1100 | . . . . . . 7 |
30 | 29 | opeq2d 3556 | . . . . . 6 |
31 | 30 | mpt2eq3dva 5569 | . . . . 5 |
32 | freceq1 5979 | . . . . 5 frec frec | |
33 | 31, 32 | syl 14 | . . . 4 frec frec |
34 | 1, 33 | syl5eq 2084 | . . 3 frec |
35 | 34 | rneqd 4563 | . 2 frec |
36 | df-iseq 9212 | . 2 frec | |
37 | 35, 36 | syl6reqr 2091 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 cop 3378 crn 4346 cfv 4902 (class class class)co 5512 cmpt2 5514 freccfrec 5977 c1 6890 caddc 6892 cuz 8473 cseq 9211 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-iseq 9212 |
This theorem is referenced by: iseqfn 9221 iseq1 9222 iseqcl 9223 iseqp1 9225 |
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