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Mirrors > Home > ILE Home > Th. List > nnind | Unicode version |
Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 7934 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Ref | Expression |
---|---|
nnind.1 | |
nnind.2 | |
nnind.3 | |
nnind.4 | |
nnind.5 | |
nnind.6 |
Ref | Expression |
---|---|
nnind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 7925 | . . . . . 6 | |
2 | nnind.5 | . . . . . 6 | |
3 | nnind.1 | . . . . . . 7 | |
4 | 3 | elrab 2698 | . . . . . 6 |
5 | 1, 2, 4 | mpbir2an 849 | . . . . 5 |
6 | elrabi 2695 | . . . . . . 7 | |
7 | peano2nn 7926 | . . . . . . . . . 10 | |
8 | 7 | a1d 22 | . . . . . . . . 9 |
9 | nnind.6 | . . . . . . . . 9 | |
10 | 8, 9 | anim12d 318 | . . . . . . . 8 |
11 | nnind.2 | . . . . . . . . 9 | |
12 | 11 | elrab 2698 | . . . . . . . 8 |
13 | nnind.3 | . . . . . . . . 9 | |
14 | 13 | elrab 2698 | . . . . . . . 8 |
15 | 10, 12, 14 | 3imtr4g 194 | . . . . . . 7 |
16 | 6, 15 | mpcom 32 | . . . . . 6 |
17 | 16 | rgen 2374 | . . . . 5 |
18 | peano5nni 7917 | . . . . 5 | |
19 | 5, 17, 18 | mp2an 402 | . . . 4 |
20 | 19 | sseli 2941 | . . 3 |
21 | nnind.4 | . . . 4 | |
22 | 21 | elrab 2698 | . . 3 |
23 | 20, 22 | sylib 127 | . 2 |
24 | 23 | simprd 107 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 crab 2310 wss 2917 (class class class)co 5512 c1 6890 caddc 6892 cn 7914 |
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-1re 6978 ax-addrcl 6981 |
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-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 |
This theorem is referenced by: nnindALT 7931 nn1m1nn 7932 nnaddcl 7934 nnmulcl 7935 nnge1 7937 nn1gt1 7947 nnsub 7952 zaddcllempos 8282 zaddcllemneg 8284 nneoor 8340 peano5uzti 8346 nn0ind-raph 8355 indstr 8536 qbtwnzlemshrink 9104 expivallem 9256 expcllem 9266 expap0 9285 resqrexlemover 9608 resqrexlemlo 9611 resqrexlemcalc3 9614 sqrt2irr 9878 |
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