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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 |
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iseqovex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2171 | . . 3 | |
2 | simprr 527 | . . . 4 | |
3 | simprl 526 | . . . . . 6 | |
4 | 3 | oveq1d 5868 | . . . . 5 |
5 | 4 | fveq2d 5500 | . . . 4 |
6 | 2, 5 | oveq12d 5871 | . . 3 |
7 | simprl 526 | . . 3 | |
8 | simprr 527 | . . 3 | |
9 | iseqovex.pl | . . . . . 6 | |
10 | 9 | caovclg 6005 | . . . . 5 |
11 | 10 | adantlr 474 | . . . 4 |
12 | fveq2 5496 | . . . . . 6 | |
13 | 12 | eleq1d 2239 | . . . . 5 |
14 | iseqovex.f | . . . . . . . 8 | |
15 | 14 | ralrimiva 2543 | . . . . . . 7 |
16 | fveq2 5496 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2239 | . . . . . . . 8 |
18 | 17 | cbvralv 2696 | . . . . . . 7 |
19 | 15, 18 | sylib 121 | . . . . . 6 |
20 | 19 | adantr 274 | . . . . 5 |
21 | peano2uz 9542 | . . . . . 6 | |
22 | 7, 21 | syl 14 | . . . . 5 |
23 | 13, 20, 22 | rspcdva 2839 | . . . 4 |
24 | 11, 8, 23 | caovcld 6006 | . . 3 |
25 | 1, 6, 7, 8, 24 | ovmpod 5980 | . 2 |
26 | 25, 24 | eqeltrd 2247 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 cfv 5198 (class class class)co 5853 cmpo 5855 c1 7775 caddc 7777 cuz 9487 |
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-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
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-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: seq3val 10414 seq3-1 10416 seq3p1 10418 |
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