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Mirrors > Home > ILE Home > Th. List > iseqf1olemkle | Unicode version |
Description: Lemma for seq3f1o 10412. (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 9934 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | 4 | zred 9291 | . . 3 |
6 | iseqf1olemkle.j | . . . . . . . . 9 | |
7 | f1ocnv 5429 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | f1of 5416 | . . . . . . . 8 | |
10 | 8, 9 | syl 14 | . . . . . . 7 |
11 | 10, 1 | ffvelrnd 5605 | . . . . . 6 |
12 | elfzelz 9934 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | 13 | adantr 274 | . . . 4 |
15 | 14 | zred 9291 | . . 3 |
16 | simpr 109 | . . 3 | |
17 | 5, 15, 16 | ltled 7998 | . 2 |
18 | 3 | zred 9291 | . . 3 |
19 | eqle 7971 | . . 3 | |
20 | 18, 19 | sylan 281 | . 2 |
21 | 6 | adantr 274 | . . . . 5 |
22 | 1 | adantr 274 | . . . . 5 |
23 | f1ocnvfv2 5730 | . . . . 5 | |
24 | 21, 22, 23 | syl2anc 409 | . . . 4 |
25 | fveq2 5470 | . . . . . 6 | |
26 | id 19 | . . . . . 6 | |
27 | 25, 26 | eqeq12d 2172 | . . . . 5 |
28 | iseqf1olemkle.const | . . . . . 6 ..^ | |
29 | 28 | adantr 274 | . . . . 5 ..^ |
30 | elfzuz 9930 | . . . . . . . 8 | |
31 | 11, 30 | syl 14 | . . . . . . 7 |
32 | 31 | adantr 274 | . . . . . 6 |
33 | 3 | adantr 274 | . . . . . 6 |
34 | simpr 109 | . . . . . 6 | |
35 | elfzo2 10058 | . . . . . 6 ..^ | |
36 | 32, 33, 34, 35 | syl3anbrc 1166 | . . . . 5 ..^ |
37 | 27, 29, 36 | rspcdva 2821 | . . . 4 |
38 | 24, 37 | eqtr3d 2192 | . . 3 |
39 | 38, 20 | syldan 280 | . 2 |
40 | ztri3or 9215 | . . 3 | |
41 | 3, 13, 40 | syl2anc 409 | . 2 |
42 | 17, 20, 39, 41 | mpjao3dan 1289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 962 wceq 1335 wcel 2128 wral 2435 class class class wbr 3967 ccnv 4587 wf 5168 wf1o 5171 cfv 5172 (class class class)co 5826 cr 7733 clt 7914 cle 7915 cz 9172 cuz 9444 cfz 9918 ..^cfzo 10050 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-n0 9096 df-z 9173 df-uz 9445 df-fz 9919 df-fzo 10051 |
This theorem is referenced by: iseqf1olemqk 10402 seq3f1olemqsumkj 10406 seq3f1olemqsumk 10407 |
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