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Mirrors > Home > ILE Home > Th. List > frecuzrdgrrn | 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 . (Contributed by Jim Kingdon, 28-Mar-2022.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
frecuzrdgrrn.a | |
frecuzrdgrrn.f | |
frecuzrdgrrn.2 | frec |
Ref | Expression |
---|---|
frecuzrdgrrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrrn.2 | . . 3 frec | |
2 | 1 | fveq1i 5470 | . 2 frec |
3 | frec2uz.1 | . . . . . 6 | |
4 | uzid 9454 | . . . . . 6 | |
5 | 3, 4 | syl 14 | . . . . 5 |
6 | frecuzrdgrrn.a | . . . . 5 | |
7 | opelxp 4617 | . . . . 5 | |
8 | 5, 6, 7 | sylanbrc 414 | . . . 4 |
9 | 8 | adantr 274 | . . 3 |
10 | 1st2nd2 6124 | . . . . . . 7 | |
11 | fveq2 5469 | . . . . . . . 8 | |
12 | df-ov 5828 | . . . . . . . 8 | |
13 | 11, 12 | eqtr4di 2208 | . . . . . . 7 |
14 | 10, 13 | syl 14 | . . . . . 6 |
15 | 14 | adantl 275 | . . . . 5 |
16 | xp1st 6114 | . . . . . . 7 | |
17 | 16 | adantl 275 | . . . . . 6 |
18 | xp2nd 6115 | . . . . . . 7 | |
19 | 18 | adantl 275 | . . . . . 6 |
20 | peano2uz 9495 | . . . . . . . 8 | |
21 | 17, 20 | syl 14 | . . . . . . 7 |
22 | frecuzrdgrrn.f | . . . . . . . . . 10 | |
23 | 22 | ralrimivva 2539 | . . . . . . . . 9 |
24 | 23 | ad2antrr 480 | . . . . . . . 8 |
25 | oveq1 5832 | . . . . . . . . . . 11 | |
26 | 25 | eleq1d 2226 | . . . . . . . . . 10 |
27 | oveq2 5833 | . . . . . . . . . . 11 | |
28 | 27 | eleq1d 2226 | . . . . . . . . . 10 |
29 | 26, 28 | rspc2v 2829 | . . . . . . . . 9 |
30 | 17, 19, 29 | syl2anc 409 | . . . . . . . 8 |
31 | 24, 30 | mpd 13 | . . . . . . 7 |
32 | opelxp 4617 | . . . . . . 7 | |
33 | 21, 31, 32 | sylanbrc 414 | . . . . . 6 |
34 | oveq1 5832 | . . . . . . . 8 | |
35 | 34, 25 | opeq12d 3750 | . . . . . . 7 |
36 | 27 | opeq2d 3749 | . . . . . . 7 |
37 | eqid 2157 | . . . . . . 7 | |
38 | 35, 36, 37 | ovmpog 5956 | . . . . . 6 |
39 | 17, 19, 33, 38 | syl3anc 1220 | . . . . 5 |
40 | 15, 39 | eqtrd 2190 | . . . 4 |
41 | 40, 33 | eqeltrd 2234 | . . 3 |
42 | simpr 109 | . . 3 | |
43 | 9, 41, 42 | freccl 6351 | . 2 frec |
44 | 2, 43 | eqeltrid 2244 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 cop 3563 cmpt 4026 com 4550 cxp 4585 cfv 5171 (class class class)co 5825 cmpo 5827 c1st 6087 c2nd 6088 freccfrec 6338 c1 7734 caddc 7736 cz 9168 cuz 9440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 |
This theorem is referenced by: frec2uzrdg 10312 frecuzrdgtcl 10315 frecuzrdgsuc 10317 |
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