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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 10214 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 6081 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | fveq2d 5433 | . . . . 5 |
4 | df-ov 5785 | . . . . . . 7 | |
5 | xp1st 6071 | . . . . . . . . 9 | |
6 | 5 | adantl 275 | . . . . . . . 8 |
7 | frecuzrdgrclt.t | . . . . . . . . . 10 | |
8 | 7 | sseld 3101 | . . . . . . . . 9 |
9 | xp2nd 6072 | . . . . . . . . 9 | |
10 | 8, 9 | impel 278 | . . . . . . . 8 |
11 | peano2uz 9405 | . . . . . . . . . 10 | |
12 | 6, 11 | syl 14 | . . . . . . . . 9 |
13 | frecuzrdgrclt.f | . . . . . . . . . . . 12 | |
14 | 13 | ralrimivva 2517 | . . . . . . . . . . 11 |
15 | 14 | adantr 274 | . . . . . . . . . 10 |
16 | 9 | adantl 275 | . . . . . . . . . . 11 |
17 | oveq1 5789 | . . . . . . . . . . . . 13 | |
18 | 17 | eleq1d 2209 | . . . . . . . . . . . 12 |
19 | oveq2 5790 | . . . . . . . . . . . . 13 | |
20 | 19 | eleq1d 2209 | . . . . . . . . . . . 12 |
21 | 18, 20 | rspc2v 2806 | . . . . . . . . . . 11 |
22 | 6, 16, 21 | syl2anc 409 | . . . . . . . . . 10 |
23 | 15, 22 | mpd 13 | . . . . . . . . 9 |
24 | opelxp 4577 | . . . . . . . . 9 | |
25 | 12, 23, 24 | sylanbrc 414 | . . . . . . . 8 |
26 | oveq1 5789 | . . . . . . . . . 10 | |
27 | 26, 17 | opeq12d 3721 | . . . . . . . . 9 |
28 | 19 | opeq2d 3720 | . . . . . . . . 9 |
29 | eqid 2140 | . . . . . . . . 9 | |
30 | 27, 28, 29 | ovmpog 5913 | . . . . . . . 8 |
31 | 6, 10, 25, 30 | syl3anc 1217 | . . . . . . 7 |
32 | 4, 31 | syl5eqr 2187 | . . . . . 6 |
33 | 32, 25 | eqeltrd 2217 | . . . . 5 |
34 | 3, 33 | eqeltrd 2217 | . . . 4 |
35 | 34 | ralrimiva 2508 | . . 3 |
36 | frecuzrdgrclt.c | . . . . 5 | |
37 | uzid 9364 | . . . . 5 | |
38 | 36, 37 | syl 14 | . . . 4 |
39 | frecuzrdgrclt.a | . . . 4 | |
40 | opelxp 4577 | . . . 4 | |
41 | 38, 39, 40 | sylanbrc 414 | . . 3 |
42 | frecfcl 6310 | . . 3 frec | |
43 | 35, 41, 42 | syl2anc 409 | . 2 frec |
44 | frecuzrdgrclt.r | . . 3 frec | |
45 | 44 | feq1i 5273 | . 2 frec |
46 | 43, 45 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 wral 2417 wss 3076 cop 3535 com 4512 cxp 4545 wf 5127 cfv 5131 (class class class)co 5782 cmpo 5784 c1st 6044 c2nd 6045 freccfrec 6295 c1 7645 caddc 7647 cz 9078 cuz 9350 |
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-ltadd 7760 |
This theorem depends on definitions: df-bi 116 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-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-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 |
This theorem is referenced by: frecuzrdgg 10220 frecuzrdgdomlem 10221 frecuzrdgfunlem 10223 frecuzrdgtclt 10225 frecuzrdg0t 10226 frecuzrdgsuctlem 10227 seq3val 10262 seqvalcd 10263 |
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