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Mirrors > Home > ILE Home > Th. List > frecuzrdgsuc | Unicode version |
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9401 for the description of as the mapping from to . (Contributed by Jim Kingdon, 28-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
uzrdg.s | |
uzrdg.a | |
uzrdg.f | |
uzrdg.2 | frec |
frecuzrdgfn.3 |
Ref | Expression |
---|---|
frecuzrdgsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . . . . . 7 | |
2 | 1 | adantr 270 | . . . . . 6 |
3 | frec2uz.2 | . . . . . 6 frec | |
4 | uzrdg.s | . . . . . . 7 | |
5 | 4 | adantr 270 | . . . . . 6 |
6 | uzrdg.a | . . . . . . 7 | |
7 | 6 | adantr 270 | . . . . . 6 |
8 | uzrdg.f | . . . . . . 7 | |
9 | 8 | adantlr 460 | . . . . . 6 |
10 | uzrdg.2 | . . . . . 6 frec | |
11 | peano2uz 8671 | . . . . . . 7 | |
12 | 11 | adantl 271 | . . . . . 6 |
13 | 2, 3, 5, 7, 9, 10, 12 | frecuzrdglem 9413 | . . . . 5 |
14 | frecuzrdgfn.3 | . . . . . 6 | |
15 | 14 | adantr 270 | . . . . 5 |
16 | 13, 15 | eleqtrrd 2158 | . . . 4 |
17 | 1, 3, 4, 6, 8, 10, 14 | frecuzrdgfn 9414 | . . . . . . 7 |
18 | fnfun 5016 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | funopfv 5234 | . . . . . 6 | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | 21 | adantr 270 | . . . 4 |
23 | 16, 22 | mpd 13 | . . 3 |
24 | 1, 3 | frec2uzf1od 9408 | . . . . . . . . 9 |
25 | f1ocnvdm 5441 | . . . . . . . . 9 | |
26 | 24, 25 | sylan 277 | . . . . . . . 8 |
27 | 2, 3, 26 | frec2uzsucd 9403 | . . . . . . 7 |
28 | f1ocnvfv2 5438 | . . . . . . . . 9 | |
29 | 24, 28 | sylan 277 | . . . . . . . 8 |
30 | 29 | oveq1d 5547 | . . . . . . 7 |
31 | 27, 30 | eqtrd 2113 | . . . . . 6 |
32 | peano2 4336 | . . . . . . . 8 | |
33 | 26, 32 | syl 14 | . . . . . . 7 |
34 | f1ocnvfv 5439 | . . . . . . . 8 | |
35 | 24, 34 | sylan 277 | . . . . . . 7 |
36 | 33, 35 | syldan 276 | . . . . . 6 |
37 | 31, 36 | mpd 13 | . . . . 5 |
38 | 37 | fveq2d 5202 | . . . 4 |
39 | 38 | fveq2d 5202 | . . 3 |
40 | 23, 39 | eqtrd 2113 | . 2 |
41 | zex 8360 | . . . . . . . . . . 11 | |
42 | uzssz 8638 | . . . . . . . . . . 11 | |
43 | 41, 42 | ssexi 3916 | . . . . . . . . . 10 |
44 | mpt2exga 5855 | . . . . . . . . . 10 | |
45 | 43, 44 | mpan 414 | . . . . . . . . 9 |
46 | vex 2604 | . . . . . . . . . 10 | |
47 | fvexg 5214 | . . . . . . . . . 10 | |
48 | 46, 47 | mpan2 415 | . . . . . . . . 9 |
49 | 5, 45, 48 | 3syl 17 | . . . . . . . 8 |
50 | 49 | alrimiv 1795 | . . . . . . 7 |
51 | opelxp 4392 | . . . . . . . . 9 | |
52 | 1, 6, 51 | sylanbrc 408 | . . . . . . . 8 |
53 | 52 | adantr 270 | . . . . . . 7 |
54 | frecsuc 6014 | . . . . . . 7 frec frec | |
55 | 50, 53, 26, 54 | syl3anc 1169 | . . . . . 6 frec frec |
56 | 10 | fveq1i 5199 | . . . . . 6 frec |
57 | 10 | fveq1i 5199 | . . . . . . 7 frec |
58 | 57 | fveq2i 5201 | . . . . . 6 frec |
59 | 55, 56, 58 | 3eqtr4g 2138 | . . . . 5 |
60 | 2, 3, 5, 7, 9, 10, 26 | frec2uzrdg 9411 | . . . . . . 7 |
61 | 60 | fveq2d 5202 | . . . . . 6 |
62 | df-ov 5535 | . . . . . 6 | |
63 | 61, 62 | syl6eqr 2131 | . . . . 5 |
64 | 2, 3, 26 | frec2uzuzd 9404 | . . . . . 6 |
65 | 2, 3, 5, 7, 9, 10 | frecuzrdgrrn 9410 | . . . . . . . 8 |
66 | 26, 65 | mpdan 412 | . . . . . . 7 |
67 | xp2nd 5813 | . . . . . . 7 | |
68 | 66, 67 | syl 14 | . . . . . 6 |
69 | 30, 12 | eqeltrd 2155 | . . . . . . 7 |
70 | 9 | caovclg 5673 | . . . . . . . 8 |
71 | 70, 64, 68 | caovcld 5674 | . . . . . . 7 |
72 | opexg 3983 | . . . . . . 7 | |
73 | 69, 71, 72 | syl2anc 403 | . . . . . 6 |
74 | oveq1 5539 | . . . . . . . 8 | |
75 | oveq1 5539 | . . . . . . . 8 | |
76 | 74, 75 | opeq12d 3578 | . . . . . . 7 |
77 | oveq2 5540 | . . . . . . . 8 | |
78 | 77 | opeq2d 3577 | . . . . . . 7 |
79 | oveq1 5539 | . . . . . . . . 9 | |
80 | oveq1 5539 | . . . . . . . . 9 | |
81 | 79, 80 | opeq12d 3578 | . . . . . . . 8 |
82 | oveq2 5540 | . . . . . . . . 9 | |
83 | 82 | opeq2d 3577 | . . . . . . . 8 |
84 | 81, 83 | cbvmpt2v 5604 | . . . . . . 7 |
85 | 76, 78, 84 | ovmpt2g 5655 | . . . . . 6 |
86 | 64, 68, 73, 85 | syl3anc 1169 | . . . . 5 |
87 | 59, 63, 86 | 3eqtrd 2117 | . . . 4 |
88 | 87 | fveq2d 5202 | . . 3 |
89 | op2ndg 5798 | . . . 4 | |
90 | 69, 71, 89 | syl2anc 403 | . . 3 |
91 | 88, 90 | eqtrd 2113 | . 2 |
92 | simpr 108 | . . . . . . 7 | |
93 | 2, 3, 5, 7, 9, 10, 92 | frecuzrdglem 9413 | . . . . . 6 |
94 | 93, 15 | eleqtrrd 2158 | . . . . 5 |
95 | funopfv 5234 | . . . . . . 7 | |
96 | 19, 95 | syl 14 | . . . . . 6 |
97 | 96 | adantr 270 | . . . . 5 |
98 | 94, 97 | mpd 13 | . . . 4 |
99 | 98 | eqcomd 2086 | . . 3 |
100 | 29, 99 | oveq12d 5550 | . 2 |
101 | 40, 91, 100 | 3eqtrd 2117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wcel 1433 cvv 2601 cop 3401 cmpt 3839 csuc 4120 com 4331 cxp 4361 ccnv 4362 crn 4364 wfun 4916 wfn 4917 wf1o 4921 cfv 4922 (class class class)co 5532 cmpt2 5534 c2nd 5786 freccfrec 6000 c1 6982 caddc 6984 cz 8351 cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 |
This theorem is referenced by: iseqp1 9445 |
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