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Mirrors > Home > ILE Home > Th. List > seq3p1 | Unicode version |
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.) |
Ref | Expression |
---|---|
seq3p1.m | |
seq3p1.f | |
seq3p1.pl |
Ref | Expression |
---|---|
seq3p1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3p1.m | . . 3 | |
2 | eluzel2 9485 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | fveq2 5494 | . . . . . 6 | |
5 | 4 | eleq1d 2239 | . . . . 5 |
6 | seq3p1.f | . . . . . 6 | |
7 | 6 | ralrimiva 2543 | . . . . 5 |
8 | uzid 9494 | . . . . . 6 | |
9 | 3, 8 | syl 14 | . . . . 5 |
10 | 5, 7, 9 | rspcdva 2839 | . . . 4 |
11 | ssv 3169 | . . . . 5 | |
12 | 11 | a1i 9 | . . . 4 |
13 | seq3p1.pl | . . . . 5 | |
14 | 6, 13 | iseqovex 10405 | . . . 4 |
15 | iseqvalcbv 10406 | . . . 4 frec frec | |
16 | 3, 15, 6, 13 | seq3val 10407 | . . . 4 frec |
17 | 3, 10, 12, 14, 15, 16 | frecuzrdgsuct 10373 | . . 3 |
18 | 1, 17 | mpdan 419 | . 2 |
19 | eqid 2170 | . . . . 5 | |
20 | 19, 3, 6, 13 | seqf 10410 | . . . 4 |
21 | 20, 1 | ffvelrnd 5630 | . . 3 |
22 | fveq2 5494 | . . . . . 6 | |
23 | 22 | eleq1d 2239 | . . . . 5 |
24 | peano2uz 9535 | . . . . . 6 | |
25 | 1, 24 | syl 14 | . . . . 5 |
26 | 23, 7, 25 | rspcdva 2839 | . . . 4 |
27 | 13, 21, 26 | caovcld 6004 | . . 3 |
28 | fvoveq1 5874 | . . . . 5 | |
29 | 28 | oveq2d 5867 | . . . 4 |
30 | oveq1 5858 | . . . 4 | |
31 | eqid 2170 | . . . 4 | |
32 | 29, 30, 31 | ovmpog 5985 | . . 3 |
33 | 1, 21, 27, 32 | syl3anc 1233 | . 2 |
34 | 18, 33 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 wss 3121 cop 3584 cfv 5196 (class class class)co 5851 cmpo 5853 freccfrec 6367 c1 7768 caddc 7770 cz 9205 cuz 9480 cseq 10394 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 df-uz 9481 df-seqfrec 10395 |
This theorem is referenced by: seq3clss 10416 seq3m1 10417 seq3fveq2 10418 seq3shft2 10422 ser3mono 10427 seq3split 10428 seq3caopr3 10430 seq3id3 10456 seq3id2 10458 seq3homo 10459 seq3z 10460 ser3ge0 10466 exp3vallem 10470 expp1 10476 facp1 10657 seq3coll 10770 resqrexlemfp1 10966 climserle 11301 clim2prod 11495 prodfap0 11501 prodfrecap 11502 ege2le3 11627 efgt1p2 11651 efgt1p 11652 algrp1 11993 pcmpt 12288 nninfdclemp1 12398 |
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