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Mirrors > Home > ILE Home > Th. List > nn0ind-raph | Unicode version |
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.) |
Ref | Expression |
---|---|
nn0ind-raph.1 | |
nn0ind-raph.2 | |
nn0ind-raph.3 | |
nn0ind-raph.4 | |
nn0ind-raph.5 | |
nn0ind-raph.6 |
Ref | Expression |
---|---|
nn0ind-raph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8290 | . 2 | |
2 | dfsbcq2 2818 | . . . 4 | |
3 | nfv 1461 | . . . . 5 | |
4 | nn0ind-raph.2 | . . . . 5 | |
5 | 3, 4 | sbhypf 2648 | . . . 4 |
6 | nfv 1461 | . . . . 5 | |
7 | nn0ind-raph.3 | . . . . 5 | |
8 | 6, 7 | sbhypf 2648 | . . . 4 |
9 | nfv 1461 | . . . . 5 | |
10 | nn0ind-raph.4 | . . . . 5 | |
11 | 9, 10 | sbhypf 2648 | . . . 4 |
12 | nfsbc1v 2833 | . . . . 5 | |
13 | 1ex 7114 | . . . . 5 | |
14 | c0ex 7113 | . . . . . . 7 | |
15 | 0nn0 8303 | . . . . . . . . . . . 12 | |
16 | eleq1a 2150 | . . . . . . . . . . . 12 | |
17 | 15, 16 | ax-mp 7 | . . . . . . . . . . 11 |
18 | nn0ind-raph.5 | . . . . . . . . . . . . . . 15 | |
19 | nn0ind-raph.1 | . . . . . . . . . . . . . . 15 | |
20 | 18, 19 | mpbiri 166 | . . . . . . . . . . . . . 14 |
21 | eqeq2 2090 | . . . . . . . . . . . . . . . 16 | |
22 | 21, 4 | syl6bir 162 | . . . . . . . . . . . . . . 15 |
23 | 22 | pm5.74d 180 | . . . . . . . . . . . . . 14 |
24 | 20, 23 | mpbii 146 | . . . . . . . . . . . . 13 |
25 | 24 | com12 30 | . . . . . . . . . . . 12 |
26 | 14, 25 | vtocle 2672 | . . . . . . . . . . 11 |
27 | nn0ind-raph.6 | . . . . . . . . . . 11 | |
28 | 17, 26, 27 | sylc 61 | . . . . . . . . . 10 |
29 | 28 | adantr 270 | . . . . . . . . 9 |
30 | oveq1 5539 | . . . . . . . . . . . . 13 | |
31 | 0p1e1 8153 | . . . . . . . . . . . . 13 | |
32 | 30, 31 | syl6eq 2129 | . . . . . . . . . . . 12 |
33 | 32 | eqeq2d 2092 | . . . . . . . . . . 11 |
34 | 33, 7 | syl6bir 162 | . . . . . . . . . 10 |
35 | 34 | imp 122 | . . . . . . . . 9 |
36 | 29, 35 | mpbird 165 | . . . . . . . 8 |
37 | 36 | ex 113 | . . . . . . 7 |
38 | 14, 37 | vtocle 2672 | . . . . . 6 |
39 | sbceq1a 2824 | . . . . . 6 | |
40 | 38, 39 | mpbid 145 | . . . . 5 |
41 | 12, 13, 40 | vtoclef 2671 | . . . 4 |
42 | nnnn0 8295 | . . . . 5 | |
43 | 42, 27 | syl 14 | . . . 4 |
44 | 2, 5, 8, 11, 41, 43 | nnind 8055 | . . 3 |
45 | nfv 1461 | . . . . 5 | |
46 | eqeq1 2087 | . . . . . 6 | |
47 | 19 | bicomd 139 | . . . . . . . . 9 |
48 | 47, 10 | sylan9bb 449 | . . . . . . . 8 |
49 | 18, 48 | mpbii 146 | . . . . . . 7 |
50 | 49 | ex 113 | . . . . . 6 |
51 | 46, 50 | sylbird 168 | . . . . 5 |
52 | 45, 14, 51 | vtoclef 2671 | . . . 4 |
53 | 52 | eqcoms 2084 | . . 3 |
54 | 44, 53 | jaoi 668 | . 2 |
55 | 1, 54 | sylbi 119 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 wsb 1685 wsbc 2815 (class class class)co 5532 cc0 6981 c1 6982 caddc 6984 cn 8039 cn0 8288 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 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-i2m1 7081 ax-0id 7084 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-n0 8289 |
This theorem is referenced by: (None) |
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