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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 8733 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 8724 | . . . . . 6 | |
2 | nnind.5 | . . . . . 6 | |
3 | nnind.1 | . . . . . . 7 | |
4 | 3 | elrab 2835 | . . . . . 6 |
5 | 1, 2, 4 | mpbir2an 926 | . . . . 5 |
6 | elrabi 2832 | . . . . . . 7 | |
7 | peano2nn 8725 | . . . . . . . . . 10 | |
8 | 7 | a1d 22 | . . . . . . . . 9 |
9 | nnind.6 | . . . . . . . . 9 | |
10 | 8, 9 | anim12d 333 | . . . . . . . 8 |
11 | nnind.2 | . . . . . . . . 9 | |
12 | 11 | elrab 2835 | . . . . . . . 8 |
13 | nnind.3 | . . . . . . . . 9 | |
14 | 13 | elrab 2835 | . . . . . . . 8 |
15 | 10, 12, 14 | 3imtr4g 204 | . . . . . . 7 |
16 | 6, 15 | mpcom 36 | . . . . . 6 |
17 | 16 | rgen 2483 | . . . . 5 |
18 | peano5nni 8716 | . . . . 5 | |
19 | 5, 17, 18 | mp2an 422 | . . . 4 |
20 | 19 | sseli 3088 | . . 3 |
21 | nnind.4 | . . . 4 | |
22 | 21 | elrab 2835 | . . 3 |
23 | 20, 22 | sylib 121 | . 2 |
24 | 23 | simprd 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 crab 2418 wss 3066 (class class class)co 5767 c1 7614 caddc 7616 cn 8713 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 |
This theorem is referenced by: nnindALT 8730 nn1m1nn 8731 nnaddcl 8733 nnmulcl 8734 nnge1 8736 nn1gt1 8747 nnsub 8752 zaddcllempos 9084 zaddcllemneg 9086 nneoor 9146 peano5uzti 9152 nn0ind-raph 9161 indstr 9381 exbtwnzlemshrink 10019 exp3vallem 10287 expcllem 10297 expap0 10316 seq3coll 10578 resqrexlemover 10775 resqrexlemlo 10778 resqrexlemcalc3 10781 gcdmultiple 11697 rplpwr 11704 prmind2 11790 prmdvdsexp 11815 sqrt2irr 11829 pw2dvdslemn 11832 dvexp 12833 |
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