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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 9025 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)) | |
2 | 0z 9023 | . . 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 9025 | . . . . . 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 984 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) → (𝜒 → 𝜃)) |
13 | 3, 4, 5, 6, 8, 12 | uzind 9120 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → 𝜏) |
14 | 2, 13 | mp3an1 1287 | . 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 1316 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 ≤ cle 7769 ℕ0cn0 8935 ℤcz 9012 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 |
This theorem is referenced by: zindd 9127 uzaddcl 9337 frecfzennn 10154 mulexp 10287 expadd 10290 expmul 10293 leexp1a 10303 bernneq 10367 faccl 10436 facdiv 10439 facwordi 10441 faclbnd 10442 faclbnd6 10445 facubnd 10446 bccl 10468 cjexp 10620 absexp 10806 binom 11208 bcxmas 11213 demoivreALT 11394 odd2np1lem 11481 alginv 11640 prmfac1 11742 ennnfonelemhf1o 11837 expcncf 12672 |
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