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Mirrors > Home > ILE Home > Th. List > iseqf1olemkle | Unicode version |
Description: Lemma for seq3f1o 10270. (Contributed by Jim Kingdon, 21-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemkle.n | |
iseqf1olemkle.k | |
iseqf1olemkle.j | |
iseqf1olemkle.const | ..^ |
Ref | Expression |
---|---|
iseqf1olemkle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemkle.k | . . . . . 6 | |
2 | elfzelz 9799 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | 4 | zred 9166 | . . 3 |
6 | iseqf1olemkle.j | . . . . . . . . 9 | |
7 | f1ocnv 5373 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | f1of 5360 | . . . . . . . 8 | |
10 | 8, 9 | syl 14 | . . . . . . 7 |
11 | 10, 1 | ffvelrnd 5549 | . . . . . 6 |
12 | elfzelz 9799 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | 13 | adantr 274 | . . . 4 |
15 | 14 | zred 9166 | . . 3 |
16 | simpr 109 | . . 3 | |
17 | 5, 15, 16 | ltled 7874 | . 2 |
18 | 3 | zred 9166 | . . 3 |
19 | eqle 7848 | . . 3 | |
20 | 18, 19 | sylan 281 | . 2 |
21 | 6 | adantr 274 | . . . . 5 |
22 | 1 | adantr 274 | . . . . 5 |
23 | f1ocnvfv2 5672 | . . . . 5 | |
24 | 21, 22, 23 | syl2anc 408 | . . . 4 |
25 | fveq2 5414 | . . . . . 6 | |
26 | id 19 | . . . . . 6 | |
27 | 25, 26 | eqeq12d 2152 | . . . . 5 |
28 | iseqf1olemkle.const | . . . . . 6 ..^ | |
29 | 28 | adantr 274 | . . . . 5 ..^ |
30 | elfzuz 9795 | . . . . . . . 8 | |
31 | 11, 30 | syl 14 | . . . . . . 7 |
32 | 31 | adantr 274 | . . . . . 6 |
33 | 3 | adantr 274 | . . . . . 6 |
34 | simpr 109 | . . . . . 6 | |
35 | elfzo2 9920 | . . . . . 6 ..^ | |
36 | 32, 33, 34, 35 | syl3anbrc 1165 | . . . . 5 ..^ |
37 | 27, 29, 36 | rspcdva 2789 | . . . 4 |
38 | 24, 37 | eqtr3d 2172 | . . 3 |
39 | 38, 20 | syldan 280 | . 2 |
40 | ztri3or 9090 | . . 3 | |
41 | 3, 13, 40 | syl2anc 408 | . 2 |
42 | 17, 20, 39, 41 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 961 wceq 1331 wcel 1480 wral 2414 class class class wbr 3924 ccnv 4533 wf 5114 wf1o 5117 cfv 5118 (class class class)co 5767 cr 7612 clt 7793 cle 7794 cz 9047 cuz 9319 cfz 9783 ..^cfzo 9912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: iseqf1olemqk 10260 seq3f1olemqsumkj 10264 seq3f1olemqsumk 10265 |
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