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Mirrors > Home > ILE Home > Th. List > zindd | Unicode version |
Description: Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
zindd.1 | |
zindd.2 | |
zindd.3 | |
zindd.4 | |
zindd.5 | |
zindd.6 | |
zindd.7 | |
zindd.8 |
Ref | Expression |
---|---|
zindd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl 9213 | . . . . . . 7 | |
2 | elznn0nn 9196 | . . . . . . 7 | |
3 | 1, 2 | sylib 121 | . . . . . 6 |
4 | simpr 109 | . . . . . . 7 | |
5 | 4 | orim2i 751 | . . . . . 6 |
6 | 3, 5 | syl 14 | . . . . 5 |
7 | zcn 9187 | . . . . . . . 8 | |
8 | 7 | negnegd 8191 | . . . . . . 7 |
9 | 8 | eleq1d 2233 | . . . . . 6 |
10 | 9 | orbi2d 780 | . . . . 5 |
11 | 6, 10 | mpbid 146 | . . . 4 |
12 | zindd.1 | . . . . . . . 8 | |
13 | 12 | imbi2d 229 | . . . . . . 7 |
14 | zindd.2 | . . . . . . . 8 | |
15 | 14 | imbi2d 229 | . . . . . . 7 |
16 | zindd.3 | . . . . . . . 8 | |
17 | 16 | imbi2d 229 | . . . . . . 7 |
18 | zindd.4 | . . . . . . . 8 | |
19 | 18 | imbi2d 229 | . . . . . . 7 |
20 | zindd.6 | . . . . . . 7 | |
21 | zindd.7 | . . . . . . . . 9 | |
22 | 21 | com12 30 | . . . . . . . 8 |
23 | 22 | a2d 26 | . . . . . . 7 |
24 | 13, 15, 17, 19, 20, 23 | nn0ind 9296 | . . . . . 6 |
25 | 24 | com12 30 | . . . . 5 |
26 | nnnn0 9112 | . . . . . . . 8 | |
27 | 13, 15, 17, 15, 20, 23 | nn0ind 9296 | . . . . . . . 8 |
28 | 26, 27 | syl 14 | . . . . . . 7 |
29 | 28 | com12 30 | . . . . . 6 |
30 | zindd.8 | . . . . . 6 | |
31 | 29, 30 | mpdd 41 | . . . . 5 |
32 | 25, 31 | jaod 707 | . . . 4 |
33 | 11, 32 | syl5 32 | . . 3 |
34 | 33 | ralrimiv 2536 | . 2 |
35 | znegcl 9213 | . . . . 5 | |
36 | negeq 8082 | . . . . . . . . 9 | |
37 | zcn 9187 | . . . . . . . . . 10 | |
38 | 37 | negnegd 8191 | . . . . . . . . 9 |
39 | 36, 38 | sylan9eqr 2219 | . . . . . . . 8 |
40 | 39 | eqcomd 2170 | . . . . . . 7 |
41 | 40, 18 | syl 14 | . . . . . 6 |
42 | 41 | bicomd 140 | . . . . 5 |
43 | 35, 42 | rspcdv 2828 | . . . 4 |
44 | 43 | com12 30 | . . 3 |
45 | 44 | ralrimiv 2536 | . 2 |
46 | zindd.5 | . . 3 | |
47 | 46 | rspccv 2822 | . 2 |
48 | 34, 45, 47 | 3syl 17 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wral 2442 (class class class)co 5836 cr 7743 cc0 7744 c1 7745 caddc 7747 cneg 8061 cn 8848 cn0 9105 cz 9182 |
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-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-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 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 |
This theorem is referenced by: efexp 11609 pcexp 12220 |
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