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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 11242 | . . . . . 6 ⊢ 1 ∈ V | |
3 | nn1suc.1 | . . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbcie 3817 | . . . . 5 ⊢ ([1 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | mpbir 230 | . . . 4 ⊢ [1 / 𝑥]𝜑 |
6 | 1nn 12256 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
7 | eleq1 2813 | . . . . . . 7 ⊢ (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 257 | . . . . . 6 ⊢ (𝐴 = 1 → 𝐴 ∈ ℕ) |
9 | nn1suc.4 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) | |
10 | 9 | sbcieg 3814 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
12 | dfsbcq 3775 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑)) | |
13 | 11, 12 | bitr3d 280 | . . . 4 ⊢ (𝐴 = 1 → (𝜃 ↔ [1 / 𝑥]𝜑)) |
14 | 5, 13 | mpbiri 257 | . . 3 ⊢ (𝐴 = 1 → 𝜃) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃)) |
16 | ovex 7452 | . . . . . 6 ⊢ (𝑦 + 1) ∈ V | |
17 | nn1suc.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) | |
18 | 16, 17 | sbcie 3817 | . . . . 5 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒) |
19 | oveq1 7426 | . . . . . 6 ⊢ (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1)) | |
20 | 19 | sbceq1d 3778 | . . . . 5 ⊢ (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
21 | 18, 20 | bitr3id 284 | . . . 4 ⊢ (𝑦 = (𝐴 − 1) → (𝜒 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
22 | nn1suc.6 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝜒) | |
23 | 21, 22 | vtoclga 3556 | . . 3 ⊢ ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑) |
24 | nncn 12253 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 11198 | . . . . . 6 ⊢ 1 ∈ ℂ | |
26 | npcan 11501 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
27 | 24, 25, 26 | sylancl 584 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴) |
28 | 27 | sbceq1d 3778 | . . . 4 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
29 | 28, 10 | bitrd 278 | . . 3 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ 𝜃)) |
30 | 23, 29 | imbitrid 243 | . 2 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃)) |
31 | nn1m1nn 12266 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | |
32 | 15, 30, 31 | mpjaod 858 | 1 ⊢ (𝐴 ∈ ℕ → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 [wsbc 3773 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 − cmin 11476 ℕcn 12245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-nn 12246 |
This theorem is referenced by: opsqrlem6 32027 |
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