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Mirrors > Home > ILE Home > Th. List > uzind2 | Unicode version |
Description: Induction on the upper integers that start after an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.) |
Ref | Expression |
---|---|
uzind2.1 | |
uzind2.2 | |
uzind2.3 | |
uzind2.4 | |
uzind2.5 | |
uzind2.6 |
Ref | Expression |
---|---|
uzind2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zltp1le 8405 | . . 3 | |
2 | peano2z 8387 | . . . . . . 7 | |
3 | uzind2.1 | . . . . . . . . . 10 | |
4 | 3 | imbi2d 228 | . . . . . . . . 9 |
5 | uzind2.2 | . . . . . . . . . 10 | |
6 | 5 | imbi2d 228 | . . . . . . . . 9 |
7 | uzind2.3 | . . . . . . . . . 10 | |
8 | 7 | imbi2d 228 | . . . . . . . . 9 |
9 | uzind2.4 | . . . . . . . . . 10 | |
10 | 9 | imbi2d 228 | . . . . . . . . 9 |
11 | uzind2.5 | . . . . . . . . . 10 | |
12 | 11 | a1i 9 | . . . . . . . . 9 |
13 | zltp1le 8405 | . . . . . . . . . . . . . . 15 | |
14 | uzind2.6 | . . . . . . . . . . . . . . . 16 | |
15 | 14 | 3expia 1140 | . . . . . . . . . . . . . . 15 |
16 | 13, 15 | sylbird 168 | . . . . . . . . . . . . . 14 |
17 | 16 | ex 113 | . . . . . . . . . . . . 13 |
18 | 17 | com3l 80 | . . . . . . . . . . . 12 |
19 | 18 | imp 122 | . . . . . . . . . . 11 |
20 | 19 | 3adant1 956 | . . . . . . . . . 10 |
21 | 20 | a2d 26 | . . . . . . . . 9 |
22 | 4, 6, 8, 10, 12, 21 | uzind 8458 | . . . . . . . 8 |
23 | 22 | 3exp 1137 | . . . . . . 7 |
24 | 2, 23 | syl 14 | . . . . . 6 |
25 | 24 | com34 82 | . . . . 5 |
26 | 25 | pm2.43a 50 | . . . 4 |
27 | 26 | imp 122 | . . 3 |
28 | 1, 27 | sylbid 148 | . 2 |
29 | 28 | 3impia 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 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: (None) |
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