![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11374 for an example of its use. See nn0ind 11800 for induction on nonnegative integers and uzind 11797, uzind4 12028 for induction on an arbitrary upper set of integers. See indstr 12039 for strong induction. See also nnindALT 11371. 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 11363 | . . . . . 6 ⊢ 1 ∈ ℕ | |
2 | nnind.5 | . . . . . 6 ⊢ 𝜓 | |
3 | nnind.1 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elrab 3585 | . . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓)) |
5 | 1, 2, 4 | mpbir2an 704 | . . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
6 | elrabi 3580 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ) | |
7 | peano2nn 11364 | . . . . . . . . . 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 604 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃))) |
11 | nnind.2 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
12 | 11 | elrab 3585 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒)) |
13 | nnind.3 | . . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
14 | 13 | elrab 3585 | . . . . . . . 8 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃)) |
15 | 10, 12, 14 | 3imtr4g 288 | . . . . . . 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 3131 | . . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
18 | peano5nni 11353 | . . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}) | |
19 | 5, 17, 18 | mp2an 685 | . . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑} |
20 | 19 | sseli 3823 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
21 | nnind.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
22 | 21 | elrab 3585 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏)) |
23 | 20, 22 | sylib 210 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏)) |
24 | 23 | simprd 491 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3117 {crab 3121 ⊆ wss 3798 (class class class)co 6905 1c1 10253 + caddc 10255 ℕcn 11350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-1cn 10310 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-nn 11351 |
This theorem is referenced by: nnindALT 11371 nn1m1nn 11372 nnaddcl 11374 nnmulcl 11375 nnmulclOLD 11376 nnge1 11380 nnne0 11386 nnsub 11395 nneo 11789 peano5uzi 11794 nn0ind-raph 11805 ser1const 13151 expcllem 13165 expeq0 13184 seqcoll 13537 relexpsucnnl 14149 relexpcnv 14152 relexprelg 14155 relexpnndm 14158 relexpaddnn 14168 climcndslem2 14956 sqrt2irr 15352 gcdmultiple 15642 rplpwr 15649 prmind2 15770 prmdvdsexp 15798 eulerthlem2 15858 pcmpt 15967 prmpwdvds 15979 vdwlem10 16065 mulgnnass 17928 imasdsf1olem 22548 ovolunlem1a 23662 ovolicc2lem3 23685 voliunlem1 23716 volsup 23722 dvexp 24115 plyco 24396 dgrcolem1 24428 vieta1 24466 emcllem6 25140 bposlem5 25426 2sqlem10 25566 dchrisum0flb 25612 iuninc 29926 nnindd 30113 ofldchr 30359 nexple 30616 esumfzf 30676 rrvsum 31062 subfacp1lem6 31713 cvmliftlem10 31822 bcprod 32166 faclimlem1 32171 incsequz 34086 bfplem1 34163 nnadd1com 38053 nnaddcom 38054 2nn0ind 38353 expmordi 38355 relexpxpnnidm 38836 relexpss1d 38838 iunrelexpmin1 38841 relexpmulnn 38842 trclrelexplem 38844 iunrelexpmin2 38845 relexp0a 38849 cotrcltrcl 38858 trclimalb2 38859 cotrclrcl 38875 inductionexd 39293 fmuldfeq 40610 dvnmptconst 40951 stoweidlem20 41031 wallispilem4 41079 wallispi2lem1 41082 wallispi2lem2 41083 dirkertrigeqlem1 41109 iccelpart 42257 nn0sumshdiglem2 43263 |
Copyright terms: Public domain | W3C validator |