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Mirrors > Home > ILE Home > Th. List > uzind | Unicode version |
Description: Induction on the upper integers that start at . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.) |
Ref | Expression |
---|---|
uzind.1 | |
uzind.2 | |
uzind.3 | |
uzind.4 | |
uzind.5 | |
uzind.6 |
Ref | Expression |
---|---|
uzind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8355 | . . . . . . . . . . 11 | |
2 | 1 | leidd 7615 | . . . . . . . . . 10 |
3 | uzind.5 | . . . . . . . . . 10 | |
4 | 2, 3 | jca 300 | . . . . . . . . 9 |
5 | 4 | ancli 316 | . . . . . . . 8 |
6 | breq2 3789 | . . . . . . . . . 10 | |
7 | uzind.1 | . . . . . . . . . 10 | |
8 | 6, 7 | anbi12d 456 | . . . . . . . . 9 |
9 | 8 | elrab 2749 | . . . . . . . 8 |
10 | 5, 9 | sylibr 132 | . . . . . . 7 |
11 | peano2z 8387 | . . . . . . . . . . . 12 | |
12 | 11 | a1i 9 | . . . . . . . . . . 11 |
13 | 12 | adantrd 273 | . . . . . . . . . 10 |
14 | zre 8355 | . . . . . . . . . . . . . 14 | |
15 | ltp1 7922 | . . . . . . . . . . . . . . . . 17 | |
16 | 15 | adantl 271 | . . . . . . . . . . . . . . . 16 |
17 | peano2re 7244 | . . . . . . . . . . . . . . . . . 18 | |
18 | 17 | ancli 316 | . . . . . . . . . . . . . . . . 17 |
19 | lelttr 7199 | . . . . . . . . . . . . . . . . . 18 | |
20 | 19 | 3expb 1139 | . . . . . . . . . . . . . . . . 17 |
21 | 18, 20 | sylan2 280 | . . . . . . . . . . . . . . . 16 |
22 | 16, 21 | mpan2d 418 | . . . . . . . . . . . . . . 15 |
23 | ltle 7198 | . . . . . . . . . . . . . . . 16 | |
24 | 17, 23 | sylan2 280 | . . . . . . . . . . . . . . 15 |
25 | 22, 24 | syld 44 | . . . . . . . . . . . . . 14 |
26 | 1, 14, 25 | syl2an 283 | . . . . . . . . . . . . 13 |
27 | 26 | adantrd 273 | . . . . . . . . . . . 12 |
28 | 27 | expimpd 355 | . . . . . . . . . . 11 |
29 | uzind.6 | . . . . . . . . . . . . 13 | |
30 | 29 | 3exp 1137 | . . . . . . . . . . . 12 |
31 | 30 | imp4d 344 | . . . . . . . . . . 11 |
32 | 28, 31 | jcad 301 | . . . . . . . . . 10 |
33 | 13, 32 | jcad 301 | . . . . . . . . 9 |
34 | breq2 3789 | . . . . . . . . . . 11 | |
35 | uzind.2 | . . . . . . . . . . 11 | |
36 | 34, 35 | anbi12d 456 | . . . . . . . . . 10 |
37 | 36 | elrab 2749 | . . . . . . . . 9 |
38 | breq2 3789 | . . . . . . . . . . 11 | |
39 | uzind.3 | . . . . . . . . . . 11 | |
40 | 38, 39 | anbi12d 456 | . . . . . . . . . 10 |
41 | 40 | elrab 2749 | . . . . . . . . 9 |
42 | 33, 37, 41 | 3imtr4g 203 | . . . . . . . 8 |
43 | 42 | ralrimiv 2433 | . . . . . . 7 |
44 | peano5uzti 8455 | . . . . . . 7 | |
45 | 10, 43, 44 | mp2and 423 | . . . . . 6 |
46 | 45 | sseld 2998 | . . . . 5 |
47 | breq2 3789 | . . . . . 6 | |
48 | 47 | elrab 2749 | . . . . 5 |
49 | breq2 3789 | . . . . . . 7 | |
50 | uzind.4 | . . . . . . 7 | |
51 | 49, 50 | anbi12d 456 | . . . . . 6 |
52 | 51 | elrab 2749 | . . . . 5 |
53 | 46, 48, 52 | 3imtr3g 202 | . . . 4 |
54 | 53 | 3impib 1136 | . . 3 |
55 | 54 | simprd 112 | . 2 |
56 | 55 | simprd 112 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wral 2348 crab 2352 wss 2973 class class class wbr 3785 (class class class)co 5532 cr 6980 c1 6982 caddc 6984 clt 7153 cle 7154 cz 8351 |
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-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 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-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 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 |
This theorem is referenced by: uzind2 8459 uzind3 8460 nn0ind 8461 fzind 8462 resqrexlemdecn 9898 ialgcvga 10433 |
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