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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 10324 | . . . . . 6 ⊢ 1 ∈ V | |
3 | nn1suc.1 | . . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbcie 3668 | . . . . 5 ⊢ ([1 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | mpbir 223 | . . . 4 ⊢ [1 / 𝑥]𝜑 |
6 | 1nn 11325 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
7 | eleq1 2866 | . . . . . . 7 ⊢ (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 250 | . . . . . 6 ⊢ (𝐴 = 1 → 𝐴 ∈ ℕ) |
9 | nn1suc.4 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) | |
10 | 9 | sbcieg 3666 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
12 | dfsbcq 3635 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑)) | |
13 | 11, 12 | bitr3d 273 | . . . 4 ⊢ (𝐴 = 1 → (𝜃 ↔ [1 / 𝑥]𝜑)) |
14 | 5, 13 | mpbiri 250 | . . 3 ⊢ (𝐴 = 1 → 𝜃) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃)) |
16 | ovex 6910 | . . . . . 6 ⊢ (𝑦 + 1) ∈ V | |
17 | nn1suc.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) | |
18 | 16, 17 | sbcie 3668 | . . . . 5 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒) |
19 | oveq1 6885 | . . . . . 6 ⊢ (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1)) | |
20 | 19 | sbceq1d 3638 | . . . . 5 ⊢ (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
21 | 18, 20 | syl5bbr 277 | . . . 4 ⊢ (𝑦 = (𝐴 − 1) → (𝜒 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
22 | nn1suc.6 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝜒) | |
23 | 21, 22 | vtoclga 3460 | . . 3 ⊢ ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑) |
24 | nncn 11321 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 10282 | . . . . . 6 ⊢ 1 ∈ ℂ | |
26 | npcan 10582 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
27 | 24, 25, 26 | sylancl 581 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴) |
28 | 27 | sbceq1d 3638 | . . . 4 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
29 | 28, 10 | bitrd 271 | . . 3 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ 𝜃)) |
30 | 23, 29 | syl5ib 236 | . 2 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃)) |
31 | nn1m1nn 11334 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | |
32 | 15, 30, 31 | mpjaod 887 | 1 ⊢ (𝐴 ∈ ℕ → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 [wsbc 3633 (class class class)co 6878 ℂcc 10222 1c1 10225 + caddc 10227 − cmin 10556 ℕcn 11312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-nn 11313 |
This theorem is referenced by: opsqrlem6 29529 |
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