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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 9175 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
2 | 0z 9173 | . . 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 9175 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)) | |
10 | nn0ind.6 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃)) | |
11 | 9, 10 | sylbir 134 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
12 | 11 | 3adant1 1000 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
13 | 3, 4, 5, 6, 8, 12 | uzind 9270 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
14 | 2, 13 | mp3an1 1306 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
15 | 1, 14 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 class class class wbr 3965 (class class class)co 5821 0cc0 7727 1c1 7728 + caddc 7730 ≤ cle 7908 ℕ0cn0 9085 ℤcz 9162 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-iota 5134 df-fun 5171 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-inn 8829 df-n0 9086 df-z 9163 |
This theorem is referenced by: zindd 9277 uzaddcl 9492 frecfzennn 10320 mulexp 10453 expadd 10456 expmul 10459 leexp1a 10469 bernneq 10533 modqexp 10539 faccl 10604 facdiv 10607 facwordi 10609 faclbnd 10610 faclbnd6 10613 facubnd 10614 bccl 10636 cjexp 10788 absexp 10974 binom 11376 bcxmas 11381 fprodfac 11507 demoivreALT 11665 odd2np1lem 11757 alginv 11918 prmfac1 12021 ennnfonelemhf1o 12129 expcncf 12979 |
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