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| Mirrors > Home > ILE Home > Th. List > nn0ind | GIF 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. (Contributed by NM, 13-May-2004.) |
| Ref | Expression |
|---|---|
| nn0ind.1 | ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) |
| nn0ind.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
| nn0ind.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
| nn0ind.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
| nn0ind.5 | ⊢ 𝜓 |
| nn0ind.6 | ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) |
| Ref | Expression |
|---|---|
| nn0ind | ⊢ (𝐴 ∈ ℕ0 → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0z 9384 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
| 2 | 0z 9382 | . . 3 ⊢ 0 ∈ ℤ | |
| 3 | nn0ind.1 | . . . 4 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
| 4 | nn0ind.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
| 5 | nn0ind.3 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
| 6 | nn0ind.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
| 7 | nn0ind.5 | . . . . 5 ⊢ 𝜓 | |
| 8 | 7 | a1i 9 | . . . 4 ⊢ (0 ∈ ℤ → 𝜓) |
| 9 | elnn0z 9384 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)) | |
| 10 | nn0ind.6 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) | |
| 11 | 9, 10 | sylbir 135 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
| 12 | 11 | 3adant1 1017 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
| 13 | 3, 4, 5, 6, 8, 12 | uzind 9483 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
| 14 | 2, 13 | mp3an1 1336 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
| 15 | 1, 14 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝜏) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1372 ∈ wcel 2175 class class class wbr 4043 (class class class)co 5943 0cc0 7924 1c1 7925 + caddc 7927 ≤ cle 8107 ℕ0cn0 9294 ℤcz 9371 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-0id 8032 ax-rnegex 8033 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-ltadd 8040 |
| This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-inn 9036 df-n0 9295 df-z 9372 |
| This theorem is referenced by: zindd 9490 uzaddcl 9706 frecfzennn 10569 mulexp 10721 expadd 10724 expmul 10727 leexp1a 10737 bernneq 10803 modqexp 10809 nn0ltexp2 10852 faccl 10878 facdiv 10881 facwordi 10883 faclbnd 10884 faclbnd6 10887 facubnd 10888 bccl 10910 cjexp 11146 absexp 11332 binom 11737 bcxmas 11742 fprodfac 11868 demoivreALT 12027 odd2np1lem 12125 bitsinv1 12215 alginv 12311 prmfac1 12416 pcfac 12615 ennnfonelemhf1o 12726 mhmmulg 13441 srgmulgass 13693 srgpcomp 13694 lmodvsmmulgdi 14027 cnfldexp 14281 expcn 14983 expcncf 15023 plycolemc 15172 rpcxpmul2 15327 |
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