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Mirrors > Home > MPE Home > Th. List > nn1suc | Structured version Visualization version GIF version |
Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1suc.1 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
nn1suc.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) |
nn1suc.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) |
nn1suc.5 | ⊢ 𝜓 |
nn1suc.6 | ⊢ (𝑦 ∈ ℕ → 𝜒) |
Ref | Expression |
---|---|
nn1suc | ⊢ (𝐴 ∈ ℕ → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1suc.5 | . . . . 5 ⊢ 𝜓 | |
2 | 1ex 10626 | . . . . . 6 ⊢ 1 ∈ V | |
3 | nn1suc.1 | . . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbcie 3760 | . . . . 5 ⊢ ([1 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | mpbir 234 | . . . 4 ⊢ [1 / 𝑥]𝜑 |
6 | 1nn 11636 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
7 | eleq1 2877 | . . . . . . 7 ⊢ (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 261 | . . . . . 6 ⊢ (𝐴 = 1 → 𝐴 ∈ ℕ) |
9 | nn1suc.4 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) | |
10 | 9 | sbcieg 3758 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
12 | dfsbcq 3722 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑)) | |
13 | 11, 12 | bitr3d 284 | . . . 4 ⊢ (𝐴 = 1 → (𝜃 ↔ [1 / 𝑥]𝜑)) |
14 | 5, 13 | mpbiri 261 | . . 3 ⊢ (𝐴 = 1 → 𝜃) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃)) |
16 | ovex 7168 | . . . . . 6 ⊢ (𝑦 + 1) ∈ V | |
17 | nn1suc.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) | |
18 | 16, 17 | sbcie 3760 | . . . . 5 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒) |
19 | oveq1 7142 | . . . . . 6 ⊢ (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1)) | |
20 | 19 | sbceq1d 3725 | . . . . 5 ⊢ (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
21 | 18, 20 | bitr3id 288 | . . . 4 ⊢ (𝑦 = (𝐴 − 1) → (𝜒 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
22 | nn1suc.6 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝜒) | |
23 | 21, 22 | vtoclga 3522 | . . 3 ⊢ ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑) |
24 | nncn 11633 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 10584 | . . . . . 6 ⊢ 1 ∈ ℂ | |
26 | npcan 10884 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
27 | 24, 25, 26 | sylancl 589 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴) |
28 | 27 | sbceq1d 3725 | . . . 4 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
29 | 28, 10 | bitrd 282 | . . 3 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ 𝜃)) |
30 | 23, 29 | syl5ib 247 | . 2 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃)) |
31 | nn1m1nn 11646 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | |
32 | 15, 30, 31 | mpjaod 857 | 1 ⊢ (𝐴 ∈ ℕ → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 [wsbc 3720 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 − cmin 10859 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-nn 11626 |
This theorem is referenced by: opsqrlem6 29928 |
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