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Mirrors > Home > ILE Home > Th. List > iseqf1olemfvp | Unicode version |
Description: Lemma for seq3f1o 10398. (Contributed by Jim Kingdon, 30-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemfvp.k | |
iseqf1olemfvp.t | |
iseqf1olemfvp.a | |
iseqf1olemfvp.g | |
iseqf1olemfvp.p |
Ref | Expression |
---|---|
iseqf1olemfvp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemfvp.p | . . . . 5 | |
2 | 1 | csbeq2i 3058 | . . . 4 |
3 | iseqf1olemfvp.t | . . . . . . 7 | |
4 | f1of 5413 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | iseqf1olemfvp.k | . . . . . . . 8 | |
7 | elfzel1 9922 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | elfzel2 9921 | . . . . . . . 8 | |
10 | 6, 9 | syl 14 | . . . . . . 7 |
11 | 8, 10 | fzfigd 10325 | . . . . . 6 |
12 | fex 5693 | . . . . . 6 | |
13 | 5, 11, 12 | syl2anc 409 | . . . . 5 |
14 | nfcvd 2300 | . . . . . 6 | |
15 | fveq1 5466 | . . . . . . . . 9 | |
16 | 15 | fveq2d 5471 | . . . . . . . 8 |
17 | 16 | ifeq1d 3522 | . . . . . . 7 |
18 | 17 | mpteq2dv 4055 | . . . . . 6 |
19 | 14, 18 | csbiegf 3074 | . . . . 5 |
20 | 13, 19 | syl 14 | . . . 4 |
21 | 2, 20 | syl5eq 2202 | . . 3 |
22 | simpr 109 | . . . . 5 | |
23 | 22 | breq1d 3975 | . . . 4 |
24 | 22 | fveq2d 5471 | . . . . 5 |
25 | 24 | fveq2d 5471 | . . . 4 |
26 | 23, 25 | ifbieq1d 3527 | . . 3 |
27 | iseqf1olemfvp.a | . . . 4 | |
28 | elfzuz 9919 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | elfzle2 9925 | . . . . . 6 | |
31 | 27, 30 | syl 14 | . . . . 5 |
32 | 31 | iftrued 3512 | . . . 4 |
33 | fveq2 5467 | . . . . . 6 | |
34 | 33 | eleq1d 2226 | . . . . 5 |
35 | iseqf1olemfvp.g | . . . . . 6 | |
36 | 35 | ralrimiva 2530 | . . . . 5 |
37 | 5, 27 | ffvelrnd 5602 | . . . . . 6 |
38 | elfzuz 9919 | . . . . . 6 | |
39 | 37, 38 | syl 14 | . . . . 5 |
40 | 34, 36, 39 | rspcdva 2821 | . . . 4 |
41 | 32, 40 | eqeltrd 2234 | . . 3 |
42 | 21, 26, 29, 41 | fvmptd 5548 | . 2 |
43 | 42, 32 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 csb 3031 cif 3505 class class class wbr 3965 cmpt 4025 wf 5165 wf1o 5168 cfv 5169 (class class class)co 5821 cfn 6682 cle 7908 cz 9162 cuz 9434 cfz 9907 |
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-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-dc 821 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-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-1o 6360 df-er 6477 df-en 6683 df-fin 6685 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-inn 8829 df-n0 9086 df-z 9163 df-uz 9435 df-fz 9908 |
This theorem is referenced by: seq3f1olemqsumkj 10392 seq3f1olemqsumk 10393 |
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