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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 10853 | . . . . . 6 ⊢ 1 ∈ V | |
3 | nn1suc.1 | . . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbcie 3751 | . . . . 5 ⊢ ([1 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | mpbir 234 | . . . 4 ⊢ [1 / 𝑥]𝜑 |
6 | 1nn 11865 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
7 | eleq1 2826 | . . . . . . 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 3748 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
12 | dfsbcq 3710 | . . . . 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 7264 | . . . . . 6 ⊢ (𝑦 + 1) ∈ V | |
17 | nn1suc.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) | |
18 | 16, 17 | sbcie 3751 | . . . . 5 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒) |
19 | oveq1 7238 | . . . . . 6 ⊢ (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1)) | |
20 | 19 | sbceq1d 3713 | . . . . 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 3501 | . . 3 ⊢ ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑) |
24 | nncn 11862 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 10811 | . . . . . 6 ⊢ 1 ∈ ℂ | |
26 | npcan 11111 | . . . . . 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 3713 | . . . 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 11875 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | |
32 | 15, 30, 31 | mpjaod 860 | 1 ⊢ (𝐴 ∈ ℕ → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2111 [wsbc 3708 (class class class)co 7231 ℂcc 10751 1c1 10754 + caddc 10756 − cmin 11086 ℕcn 11854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-ltxr 10896 df-sub 11088 df-nn 11855 |
This theorem is referenced by: opsqrlem6 30250 |
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