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Mirrors > Home > ILE Home > Th. List > uzsinds | Unicode version |
Description: Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
uzsinds.1 | |
uzsinds.2 | |
uzsinds.3 |
Ref | Expression |
---|---|
uzsinds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzsinds.2 | . 2 | |
2 | oveq2 5844 | . . . 4 | |
3 | 2 | raleqdv 2665 | . . 3 |
4 | oveq2 5844 | . . . 4 | |
5 | 4 | raleqdv 2665 | . . 3 |
6 | oveq2 5844 | . . . 4 | |
7 | 6 | raleqdv 2665 | . . 3 |
8 | oveq2 5844 | . . . 4 | |
9 | 8 | raleqdv 2665 | . . 3 |
10 | ral0 3505 | . . . . . . 7 | |
11 | zre 9186 | . . . . . . . . . 10 | |
12 | 11 | ltm1d 8818 | . . . . . . . . 9 |
13 | peano2zm 9220 | . . . . . . . . . 10 | |
14 | fzn 9967 | . . . . . . . . . 10 | |
15 | 13, 14 | mpdan 418 | . . . . . . . . 9 |
16 | 12, 15 | mpbid 146 | . . . . . . . 8 |
17 | 16 | raleqdv 2665 | . . . . . . 7 |
18 | 10, 17 | mpbiri 167 | . . . . . 6 |
19 | uzid 9471 | . . . . . . 7 | |
20 | uzsinds.3 | . . . . . . . 8 | |
21 | 20 | rgen 2517 | . . . . . . 7 |
22 | nfv 1515 | . . . . . . . . 9 | |
23 | nfsbc1v 2964 | . . . . . . . . 9 | |
24 | 22, 23 | nfim 1559 | . . . . . . . 8 |
25 | oveq1 5843 | . . . . . . . . . . 11 | |
26 | 25 | oveq2d 5852 | . . . . . . . . . 10 |
27 | 26 | raleqdv 2665 | . . . . . . . . 9 |
28 | sbceq1a 2955 | . . . . . . . . 9 | |
29 | 27, 28 | imbi12d 233 | . . . . . . . 8 |
30 | 24, 29 | rspc 2819 | . . . . . . 7 |
31 | 19, 21, 30 | mpisyl 1433 | . . . . . 6 |
32 | 18, 31 | mpd 13 | . . . . 5 |
33 | ralsns 3608 | . . . . 5 | |
34 | 32, 33 | mpbird 166 | . . . 4 |
35 | fzsn 9991 | . . . . 5 | |
36 | 35 | raleqdv 2665 | . . . 4 |
37 | 34, 36 | mpbird 166 | . . 3 |
38 | simpr 109 | . . . . . 6 | |
39 | uzsinds.1 | . . . . . . . . . 10 | |
40 | 39 | cbvralv 2689 | . . . . . . . . 9 |
41 | 38, 40 | sylib 121 | . . . . . . . 8 |
42 | eluzelz 9466 | . . . . . . . . . . . . . 14 | |
43 | 42 | adantr 274 | . . . . . . . . . . . . 13 |
44 | 43 | zcnd 9305 | . . . . . . . . . . . 12 |
45 | 1cnd 7906 | . . . . . . . . . . . 12 | |
46 | 44, 45 | pncand 8201 | . . . . . . . . . . 11 |
47 | 46 | oveq2d 5852 | . . . . . . . . . 10 |
48 | 47 | raleqdv 2665 | . . . . . . . . 9 |
49 | peano2uz 9512 | . . . . . . . . . . 11 | |
50 | 49 | adantr 274 | . . . . . . . . . 10 |
51 | nfv 1515 | . . . . . . . . . . . 12 | |
52 | nfsbc1v 2964 | . . . . . . . . . . . 12 | |
53 | 51, 52 | nfim 1559 | . . . . . . . . . . 11 |
54 | oveq1 5843 | . . . . . . . . . . . . . 14 | |
55 | 54 | oveq2d 5852 | . . . . . . . . . . . . 13 |
56 | 55 | raleqdv 2665 | . . . . . . . . . . . 12 |
57 | sbceq1a 2955 | . . . . . . . . . . . 12 | |
58 | 56, 57 | imbi12d 233 | . . . . . . . . . . 11 |
59 | 53, 58 | rspc 2819 | . . . . . . . . . 10 |
60 | 50, 21, 59 | mpisyl 1433 | . . . . . . . . 9 |
61 | 48, 60 | sylbird 169 | . . . . . . . 8 |
62 | 41, 61 | mpd 13 | . . . . . . 7 |
63 | 42 | peano2zd 9307 | . . . . . . . . 9 |
64 | 63 | adantr 274 | . . . . . . . 8 |
65 | ralsns 3608 | . . . . . . . 8 | |
66 | 64, 65 | syl 14 | . . . . . . 7 |
67 | 62, 66 | mpbird 166 | . . . . . 6 |
68 | ralun 3299 | . . . . . 6 | |
69 | 38, 67, 68 | syl2anc 409 | . . . . 5 |
70 | fzsuc 9994 | . . . . . . 7 | |
71 | 70 | raleqdv 2665 | . . . . . 6 |
72 | 71 | adantr 274 | . . . . 5 |
73 | 69, 72 | mpbird 166 | . . . 4 |
74 | 73 | ex 114 | . . 3 |
75 | 3, 5, 7, 9, 37, 74 | uzind4 9517 | . 2 |
76 | eluzfz2 9957 | . 2 | |
77 | 1, 75, 76 | rspcdva 2830 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wsbc 2946 cun 3109 c0 3404 csn 3570 class class class wbr 3976 cfv 5182 (class class class)co 5836 c1 7745 caddc 7747 clt 7924 cmin 8060 cz 9182 cuz 9457 cfz 9935 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: nnsinds 10368 nn0sinds 10369 |
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