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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 9060 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
2 | 0z 9058 | . . 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 9060 | . . . . . 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 999 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
13 | 3, 4, 5, 6, 8, 12 | uzind 9155 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
14 | 2, 13 | mp3an1 1302 | . 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 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 ≤ cle 7794 ℕ0cn0 8970 ℤcz 9047 |
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 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: zindd 9162 uzaddcl 9374 frecfzennn 10192 mulexp 10325 expadd 10328 expmul 10331 leexp1a 10341 bernneq 10405 faccl 10474 facdiv 10477 facwordi 10479 faclbnd 10480 faclbnd6 10483 facubnd 10484 bccl 10506 cjexp 10658 absexp 10844 binom 11246 bcxmas 11251 demoivreALT 11469 odd2np1lem 11558 alginv 11717 prmfac1 11819 ennnfonelemhf1o 11915 expcncf 12750 |
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