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Mirrors > Home > MPE Home > Th. List > nnind | Structured version Visualization version GIF 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 11648 for an example of its use. See nn0ind 12065 for induction on nonnegative integers and uzind 12062, uzind4 12294 for induction on an arbitrary upper set of integers. See indstr 12304 for strong induction. See also nnindALT 11644. 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 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
nnind.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
nnind.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
nnind.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
nnind.5 | ⊢ 𝜓 |
nnind.6 | ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
nnind | ⊢ (𝐴 ∈ ℕ → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
2 | nnind.5 | . . . . . 6 ⊢ 𝜓 | |
3 | nnind.1 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elrab 3628 | . . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓)) |
5 | 1, 2, 4 | mpbir2an 710 | . . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
6 | elrabi 3623 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ) | |
7 | peano2nn 11637 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ) | |
8 | 7 | a1d 25 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)) |
9 | nnind.6 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) | |
10 | 8, 9 | anim12d 611 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃))) |
11 | nnind.2 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
12 | 11 | elrab 3628 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒)) |
13 | nnind.3 | . . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
14 | 13 | elrab 3628 | . . . . . . . 8 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃)) |
15 | 10, 12, 14 | 3imtr4g 299 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
16 | 6, 15 | mpcom 38 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
17 | 16 | rgen 3116 | . . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
18 | peano5nni 11628 | . . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}) | |
19 | 5, 17, 18 | mp2an 691 | . . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑} |
20 | 19 | sseli 3911 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
21 | nnind.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
22 | 21 | elrab 3628 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏)) |
23 | 20, 22 | sylib 221 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏)) |
24 | 23 | simprd 499 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 (class class class)co 7135 1c1 10527 + caddc 10529 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 |
This theorem is referenced by: nnindALT 11644 nnindd 11645 nn1m1nn 11646 nnaddcl 11648 nnmulcl 11649 nnge1 11653 nnne0 11659 nnsub 11669 nneo 12054 peano5uzi 12059 nn0ind-raph 12070 ser1const 13422 expcllem 13436 expeq0 13455 expmordi 13527 seqcoll 13818 relexpsucnnl 14381 relexpcnv 14386 relexprelg 14389 relexpnndm 14392 relexpaddnn 14402 climcndslem2 15197 sqrt2irr 15594 gcdmultipleOLD 15890 rplpwr 15897 prmind2 16019 prmdvdsexp 16049 eulerthlem2 16109 pcmpt 16218 prmpwdvds 16230 vdwlem10 16316 mulgnnass 18254 imasdsf1olem 22980 ovolunlem1a 24100 ovolicc2lem3 24123 voliunlem1 24154 volsup 24160 dvexp 24556 plyco 24838 dgrcolem1 24870 vieta1 24908 emcllem6 25586 bposlem5 25872 2sqlem10 26012 dchrisum0flb 26094 iuninc 30324 ofldchr 30938 nexple 31378 esumfzf 31438 rrvsum 31822 subfacp1lem6 32545 cvmliftlem10 32654 bcprod 33083 faclimlem1 33088 incsequz 35186 bfplem1 35260 nnn1suc 39467 nnadd1com 39468 nnaddcom 39469 nnadddir 39471 nnmul1com 39472 nnmulcom 39473 2nn0ind 39886 relexpxpnnidm 40404 relexpss1d 40406 iunrelexpmin1 40409 relexpmulnn 40410 trclrelexplem 40412 iunrelexpmin2 40413 relexp0a 40417 cotrcltrcl 40426 trclimalb2 40427 cotrclrcl 40443 inductionexd 40858 fmuldfeq 42225 dvnmptconst 42583 stoweidlem20 42662 wallispilem4 42710 wallispi2lem1 42713 wallispi2lem2 42714 dirkertrigeqlem1 42740 iccelpart 43950 nn0sumshdiglem2 45036 |
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