Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > frecuzrdgrcl | Unicode version |
Description: The function (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
frecuzrdgrrn.a | |
frecuzrdgrrn.f | |
frecuzrdgrrn.2 | frec |
Ref | Expression |
---|---|
frecuzrdgrcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 6041 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | fveq2d 5393 | . . . . 5 |
4 | df-ov 5745 | . . . . . . 7 | |
5 | xp1st 6031 | . . . . . . . . 9 | |
6 | 5 | adantl 275 | . . . . . . . 8 |
7 | xp2nd 6032 | . . . . . . . . 9 | |
8 | 7 | adantl 275 | . . . . . . . 8 |
9 | peano2uz 9334 | . . . . . . . . . 10 | |
10 | 6, 9 | syl 14 | . . . . . . . . 9 |
11 | frecuzrdgrrn.f | . . . . . . . . . . . 12 | |
12 | 11 | ralrimivva 2491 | . . . . . . . . . . 11 |
13 | 12 | adantr 274 | . . . . . . . . . 10 |
14 | oveq1 5749 | . . . . . . . . . . . . 13 | |
15 | 14 | eleq1d 2186 | . . . . . . . . . . . 12 |
16 | oveq2 5750 | . . . . . . . . . . . . 13 | |
17 | 16 | eleq1d 2186 | . . . . . . . . . . . 12 |
18 | 15, 17 | rspc2v 2776 | . . . . . . . . . . 11 |
19 | 6, 8, 18 | syl2anc 408 | . . . . . . . . . 10 |
20 | 13, 19 | mpd 13 | . . . . . . . . 9 |
21 | opelxp 4539 | . . . . . . . . 9 | |
22 | 10, 20, 21 | sylanbrc 413 | . . . . . . . 8 |
23 | oveq1 5749 | . . . . . . . . . 10 | |
24 | 23, 14 | opeq12d 3683 | . . . . . . . . 9 |
25 | 16 | opeq2d 3682 | . . . . . . . . 9 |
26 | eqid 2117 | . . . . . . . . 9 | |
27 | 24, 25, 26 | ovmpog 5873 | . . . . . . . 8 |
28 | 6, 8, 22, 27 | syl3anc 1201 | . . . . . . 7 |
29 | 4, 28 | syl5eqr 2164 | . . . . . 6 |
30 | 29, 22 | eqeltrd 2194 | . . . . 5 |
31 | 3, 30 | eqeltrd 2194 | . . . 4 |
32 | 31 | ralrimiva 2482 | . . 3 |
33 | frec2uz.1 | . . . . 5 | |
34 | uzid 9296 | . . . . 5 | |
35 | 33, 34 | syl 14 | . . . 4 |
36 | frecuzrdgrrn.a | . . . 4 | |
37 | opelxp 4539 | . . . 4 | |
38 | 35, 36, 37 | sylanbrc 413 | . . 3 |
39 | frecfcl 6270 | . . 3 frec | |
40 | 32, 38, 39 | syl2anc 408 | . 2 frec |
41 | frecuzrdgrrn.2 | . . 3 frec | |
42 | 41 | feq1i 5235 | . 2 frec |
43 | 40, 42 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wral 2393 cop 3500 cmpt 3959 com 4474 cxp 4507 wf 5089 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 freccfrec 6255 c1 7589 caddc 7591 cz 9012 cuz 9282 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 |
This theorem is referenced by: frecuzrdglem 10139 frecuzrdgtcl 10140 frecuzrdg0 10141 |
Copyright terms: Public domain | W3C validator |