Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iseqf1olemfvp | Unicode version |
Description: Lemma for seq3f1o 10308. (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 3034 | . . . 4 |
3 | iseqf1olemfvp.t | . . . . . . 7 | |
4 | f1of 5375 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | iseqf1olemfvp.k | . . . . . . . 8 | |
7 | elfzel1 9836 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | elfzel2 9835 | . . . . . . . 8 | |
10 | 6, 9 | syl 14 | . . . . . . 7 |
11 | 8, 10 | fzfigd 10235 | . . . . . 6 |
12 | fex 5655 | . . . . . 6 | |
13 | 5, 11, 12 | syl2anc 409 | . . . . 5 |
14 | nfcvd 2283 | . . . . . 6 | |
15 | fveq1 5428 | . . . . . . . . 9 | |
16 | 15 | fveq2d 5433 | . . . . . . . 8 |
17 | 16 | ifeq1d 3494 | . . . . . . 7 |
18 | 17 | mpteq2dv 4027 | . . . . . 6 |
19 | 14, 18 | csbiegf 3048 | . . . . 5 |
20 | 13, 19 | syl 14 | . . . 4 |
21 | 2, 20 | syl5eq 2185 | . . 3 |
22 | simpr 109 | . . . . 5 | |
23 | 22 | breq1d 3947 | . . . 4 |
24 | 22 | fveq2d 5433 | . . . . 5 |
25 | 24 | fveq2d 5433 | . . . 4 |
26 | 23, 25 | ifbieq1d 3499 | . . 3 |
27 | iseqf1olemfvp.a | . . . 4 | |
28 | elfzuz 9833 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | elfzle2 9839 | . . . . . 6 | |
31 | 27, 30 | syl 14 | . . . . 5 |
32 | 31 | iftrued 3486 | . . . 4 |
33 | fveq2 5429 | . . . . . 6 | |
34 | 33 | eleq1d 2209 | . . . . 5 |
35 | iseqf1olemfvp.g | . . . . . 6 | |
36 | 35 | ralrimiva 2508 | . . . . 5 |
37 | 5, 27 | ffvelrnd 5564 | . . . . . 6 |
38 | elfzuz 9833 | . . . . . 6 | |
39 | 37, 38 | syl 14 | . . . . 5 |
40 | 34, 36, 39 | rspcdva 2798 | . . . 4 |
41 | 32, 40 | eqeltrd 2217 | . . 3 |
42 | 21, 26, 29, 41 | fvmptd 5510 | . 2 |
43 | 42, 32 | eqtrd 2173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 cvv 2689 csb 3007 cif 3479 class class class wbr 3937 cmpt 3997 wf 5127 wf1o 5130 cfv 5131 (class class class)co 5782 cfn 6642 cle 7825 cz 9078 cuz 9350 cfz 9821 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-1o 6321 df-er 6437 df-en 6643 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: seq3f1olemqsumkj 10302 seq3f1olemqsumk 10303 |
Copyright terms: Public domain | W3C validator |