Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iseqovex | Unicode version |
Description: Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.) |
Ref | Expression |
---|---|
iseqovex.f | |
iseqovex.pl |
Ref | Expression |
---|---|
iseqovex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2082 | . . 3 | |
2 | simprr 498 | . . . 4 | |
3 | simprl 497 | . . . . . 6 | |
4 | 3 | oveq1d 5547 | . . . . 5 |
5 | 4 | fveq2d 5202 | . . . 4 |
6 | 2, 5 | oveq12d 5550 | . . 3 |
7 | simprl 497 | . . 3 | |
8 | simprr 498 | . . 3 | |
9 | iseqovex.pl | . . . . . 6 | |
10 | 9 | caovclg 5673 | . . . . 5 |
11 | 10 | adantlr 460 | . . . 4 |
12 | peano2uz 8671 | . . . . . 6 | |
13 | 7, 12 | syl 14 | . . . . 5 |
14 | iseqovex.f | . . . . . . . 8 | |
15 | 14 | ralrimiva 2434 | . . . . . . 7 |
16 | fveq2 5198 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2147 | . . . . . . . 8 |
18 | 17 | cbvralv 2577 | . . . . . . 7 |
19 | 15, 18 | sylib 120 | . . . . . 6 |
20 | 19 | adantr 270 | . . . . 5 |
21 | fveq2 5198 | . . . . . . 7 | |
22 | 21 | eleq1d 2147 | . . . . . 6 |
23 | 22 | rspcv 2697 | . . . . 5 |
24 | 13, 20, 23 | sylc 61 | . . . 4 |
25 | 11, 8, 24 | caovcld 5674 | . . 3 |
26 | 1, 6, 7, 8, 25 | ovmpt2d 5648 | . 2 |
27 | 26, 25 | eqeltrd 2155 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wral 2348 cfv 4922 (class class class)co 5532 cmpt2 5534 c1 6982 caddc 6984 cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 |
This theorem is referenced by: iseqfn 9441 iseq1 9442 iseqcl 9443 iseqp1 9445 |
Copyright terms: Public domain | W3C validator |