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Mirrors > Home > ILE Home > Th. List > indstr | Unicode version |
Description: Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.) |
Ref | Expression |
---|---|
indstr.1 | |
indstr.2 |
Ref | Expression |
---|---|
indstr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3903 | . . . . 5 | |
2 | 1 | imbi1d 230 | . . . 4 |
3 | 2 | ralbidv 2414 | . . 3 |
4 | breq2 3903 | . . . . 5 | |
5 | 4 | imbi1d 230 | . . . 4 |
6 | 5 | ralbidv 2414 | . . 3 |
7 | breq2 3903 | . . . . 5 | |
8 | 7 | imbi1d 230 | . . . 4 |
9 | 8 | ralbidv 2414 | . . 3 |
10 | breq2 3903 | . . . . 5 | |
11 | 10 | imbi1d 230 | . . . 4 |
12 | 11 | ralbidv 2414 | . . 3 |
13 | nnnlt1 8714 | . . . . 5 | |
14 | 13 | pm2.21d 593 | . . . 4 |
15 | 14 | rgen 2462 | . . 3 |
16 | 1nn 8699 | . . . . 5 | |
17 | elex2 2676 | . . . . 5 | |
18 | nfra1 2443 | . . . . . 6 | |
19 | 18 | r19.3rm 3421 | . . . . 5 |
20 | 16, 17, 19 | mp2b 8 | . . . 4 |
21 | rsp 2457 | . . . . . . . . . 10 | |
22 | 21 | com12 30 | . . . . . . . . 9 |
23 | 22 | adantl 275 | . . . . . . . 8 |
24 | indstr.2 | . . . . . . . . . . . . 13 | |
25 | 24 | rgen 2462 | . . . . . . . . . . . 12 |
26 | nfv 1493 | . . . . . . . . . . . . 13 | |
27 | nfv 1493 | . . . . . . . . . . . . . 14 | |
28 | nfsbc1v 2900 | . . . . . . . . . . . . . 14 | |
29 | 27, 28 | nfim 1536 | . . . . . . . . . . . . 13 |
30 | breq2 3903 | . . . . . . . . . . . . . . . 16 | |
31 | 30 | imbi1d 230 | . . . . . . . . . . . . . . 15 |
32 | 31 | ralbidv 2414 | . . . . . . . . . . . . . 14 |
33 | sbceq1a 2891 | . . . . . . . . . . . . . 14 | |
34 | 32, 33 | imbi12d 233 | . . . . . . . . . . . . 13 |
35 | 26, 29, 34 | cbvral 2627 | . . . . . . . . . . . 12 |
36 | 25, 35 | mpbi 144 | . . . . . . . . . . 11 |
37 | 36 | rspec 2461 | . . . . . . . . . 10 |
38 | vex 2663 | . . . . . . . . . . . . 13 | |
39 | indstr.1 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | sbcie 2915 | . . . . . . . . . . . 12 |
41 | dfsbcq 2884 | . . . . . . . . . . . 12 | |
42 | 40, 41 | syl5bbr 193 | . . . . . . . . . . 11 |
43 | 42 | biimprcd 159 | . . . . . . . . . 10 |
44 | 37, 43 | syl6 33 | . . . . . . . . 9 |
45 | 44 | adantr 274 | . . . . . . . 8 |
46 | 23, 45 | jcad 305 | . . . . . . 7 |
47 | jaob 684 | . . . . . . 7 | |
48 | 46, 47 | syl6ibr 161 | . . . . . 6 |
49 | nnleltp1 9081 | . . . . . . . . 9 | |
50 | nnz 9041 | . . . . . . . . . 10 | |
51 | nnz 9041 | . . . . . . . . . 10 | |
52 | zleloe 9069 | . . . . . . . . . 10 | |
53 | 50, 51, 52 | syl2an 287 | . . . . . . . . 9 |
54 | 49, 53 | bitr3d 189 | . . . . . . . 8 |
55 | 54 | ancoms 266 | . . . . . . 7 |
56 | 55 | imbi1d 230 | . . . . . 6 |
57 | 48, 56 | sylibrd 168 | . . . . 5 |
58 | 57 | ralimdva 2476 | . . . 4 |
59 | 20, 58 | syl5bi 151 | . . 3 |
60 | 3, 6, 9, 12, 15, 59 | nnind 8704 | . 2 |
61 | 60, 24 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wex 1453 wcel 1465 wral 2393 wsbc 2882 class class class wbr 3899 (class class class)co 5742 c1 7589 caddc 7591 clt 7768 cle 7769 cn 8688 cz 9022 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: indstr2 9371 |
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