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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 8183 | . 2 | |
2 | dfsbcq2 2767 | . . . 4 | |
3 | nfv 1421 | . . . . 5 | |
4 | nn0ind-raph.2 | . . . . 5 | |
5 | 3, 4 | sbhypf 2603 | . . . 4 |
6 | nfv 1421 | . . . . 5 | |
7 | nn0ind-raph.3 | . . . . 5 | |
8 | 6, 7 | sbhypf 2603 | . . . 4 |
9 | nfv 1421 | . . . . 5 | |
10 | nn0ind-raph.4 | . . . . 5 | |
11 | 9, 10 | sbhypf 2603 | . . . 4 |
12 | nfsbc1v 2782 | . . . . 5 | |
13 | 1ex 7022 | . . . . 5 | |
14 | c0ex 7021 | . . . . . . 7 | |
15 | 0nn0 8196 | . . . . . . . . . . . 12 | |
16 | eleq1a 2109 | . . . . . . . . . . . 12 | |
17 | 15, 16 | ax-mp 7 | . . . . . . . . . . 11 |
18 | nn0ind-raph.5 | . . . . . . . . . . . . . . 15 | |
19 | nn0ind-raph.1 | . . . . . . . . . . . . . . 15 | |
20 | 18, 19 | mpbiri 157 | . . . . . . . . . . . . . 14 |
21 | eqeq2 2049 | . . . . . . . . . . . . . . . 16 | |
22 | 21, 4 | syl6bir 153 | . . . . . . . . . . . . . . 15 |
23 | 22 | pm5.74d 171 | . . . . . . . . . . . . . 14 |
24 | 20, 23 | mpbii 136 | . . . . . . . . . . . . 13 |
25 | 24 | com12 27 | . . . . . . . . . . . 12 |
26 | 14, 25 | vtocle 2627 | . . . . . . . . . . 11 |
27 | nn0ind-raph.6 | . . . . . . . . . . 11 | |
28 | 17, 26, 27 | sylc 56 | . . . . . . . . . 10 |
29 | 28 | adantr 261 | . . . . . . . . 9 |
30 | oveq1 5519 | . . . . . . . . . . . . 13 | |
31 | 0p1e1 8031 | . . . . . . . . . . . . 13 | |
32 | 30, 31 | syl6eq 2088 | . . . . . . . . . . . 12 |
33 | 32 | eqeq2d 2051 | . . . . . . . . . . 11 |
34 | 33, 7 | syl6bir 153 | . . . . . . . . . 10 |
35 | 34 | imp 115 | . . . . . . . . 9 |
36 | 29, 35 | mpbird 156 | . . . . . . . 8 |
37 | 36 | ex 108 | . . . . . . 7 |
38 | 14, 37 | vtocle 2627 | . . . . . 6 |
39 | sbceq1a 2773 | . . . . . 6 | |
40 | 38, 39 | mpbid 135 | . . . . 5 |
41 | 12, 13, 40 | vtoclef 2626 | . . . 4 |
42 | nnnn0 8188 | . . . . 5 | |
43 | 42, 27 | syl 14 | . . . 4 |
44 | 2, 5, 8, 11, 41, 43 | nnind 7930 | . . 3 |
45 | nfv 1421 | . . . . 5 | |
46 | eqeq1 2046 | . . . . . 6 | |
47 | 19 | bicomd 129 | . . . . . . . . 9 |
48 | 47, 10 | sylan9bb 435 | . . . . . . . 8 |
49 | 18, 48 | mpbii 136 | . . . . . . 7 |
50 | 49 | ex 108 | . . . . . 6 |
51 | 46, 50 | sylbird 159 | . . . . 5 |
52 | 45, 14, 51 | vtoclef 2626 | . . . 4 |
53 | 52 | eqcoms 2043 | . . 3 |
54 | 44, 53 | jaoi 636 | . 2 |
55 | 1, 54 | sylbi 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wsb 1645 wsbc 2764 (class class class)co 5512 cc0 6889 c1 6890 caddc 6892 cn 7914 cn0 8181 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-i2m1 6989 ax-0id 6992 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-inn 7915 df-n0 8182 |
This theorem is referenced by: (None) |
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