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Mirrors > Home > ILE Home > Th. List > frecuzrdgrclt | Unicode version |
Description: The function (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of . Similar to frecuzrdgrcl 10183 except that and need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | |
frecuzrdgrclt.a | |
frecuzrdgrclt.t | |
frecuzrdgrclt.f | |
frecuzrdgrclt.r | frec |
Ref | Expression |
---|---|
frecuzrdgrclt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 6073 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | fveq2d 5425 | . . . . 5 |
4 | df-ov 5777 | . . . . . . 7 | |
5 | xp1st 6063 | . . . . . . . . 9 | |
6 | 5 | adantl 275 | . . . . . . . 8 |
7 | frecuzrdgrclt.t | . . . . . . . . . 10 | |
8 | 7 | sseld 3096 | . . . . . . . . 9 |
9 | xp2nd 6064 | . . . . . . . . 9 | |
10 | 8, 9 | impel 278 | . . . . . . . 8 |
11 | peano2uz 9378 | . . . . . . . . . 10 | |
12 | 6, 11 | syl 14 | . . . . . . . . 9 |
13 | frecuzrdgrclt.f | . . . . . . . . . . . 12 | |
14 | 13 | ralrimivva 2514 | . . . . . . . . . . 11 |
15 | 14 | adantr 274 | . . . . . . . . . 10 |
16 | 9 | adantl 275 | . . . . . . . . . . 11 |
17 | oveq1 5781 | . . . . . . . . . . . . 13 | |
18 | 17 | eleq1d 2208 | . . . . . . . . . . . 12 |
19 | oveq2 5782 | . . . . . . . . . . . . 13 | |
20 | 19 | eleq1d 2208 | . . . . . . . . . . . 12 |
21 | 18, 20 | rspc2v 2802 | . . . . . . . . . . 11 |
22 | 6, 16, 21 | syl2anc 408 | . . . . . . . . . 10 |
23 | 15, 22 | mpd 13 | . . . . . . . . 9 |
24 | opelxp 4569 | . . . . . . . . 9 | |
25 | 12, 23, 24 | sylanbrc 413 | . . . . . . . 8 |
26 | oveq1 5781 | . . . . . . . . . 10 | |
27 | 26, 17 | opeq12d 3713 | . . . . . . . . 9 |
28 | 19 | opeq2d 3712 | . . . . . . . . 9 |
29 | eqid 2139 | . . . . . . . . 9 | |
30 | 27, 28, 29 | ovmpog 5905 | . . . . . . . 8 |
31 | 6, 10, 25, 30 | syl3anc 1216 | . . . . . . 7 |
32 | 4, 31 | syl5eqr 2186 | . . . . . 6 |
33 | 32, 25 | eqeltrd 2216 | . . . . 5 |
34 | 3, 33 | eqeltrd 2216 | . . . 4 |
35 | 34 | ralrimiva 2505 | . . 3 |
36 | frecuzrdgrclt.c | . . . . 5 | |
37 | uzid 9340 | . . . . 5 | |
38 | 36, 37 | syl 14 | . . . 4 |
39 | frecuzrdgrclt.a | . . . 4 | |
40 | opelxp 4569 | . . . 4 | |
41 | 38, 39, 40 | sylanbrc 413 | . . 3 |
42 | frecfcl 6302 | . . 3 frec | |
43 | 35, 41, 42 | syl2anc 408 | . 2 frec |
44 | frecuzrdgrclt.r | . . 3 frec | |
45 | 44 | feq1i 5265 | . 2 frec |
46 | 43, 45 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 wss 3071 cop 3530 com 4504 cxp 4537 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 c1st 6036 c2nd 6037 freccfrec 6287 c1 7621 caddc 7623 cz 9054 cuz 9326 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: frecuzrdgg 10189 frecuzrdgdomlem 10190 frecuzrdgfunlem 10192 frecuzrdgtclt 10194 frecuzrdg0t 10195 frecuzrdgsuctlem 10196 seq3val 10231 seqvalcd 10232 |
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