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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 10035 | . . . . . 6 ⊢ 1 ∈ V | |
3 | nn1suc.1 | . . . . . 6 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbcie 3470 | . . . . 5 ⊢ ([1 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | mpbir 221 | . . . 4 ⊢ [1 / 𝑥]𝜑 |
6 | 1nn 11031 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
7 | eleq1 2689 | . . . . . . 7 ⊢ (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ)) | |
8 | 6, 7 | mpbiri 248 | . . . . . 6 ⊢ (𝐴 = 1 → 𝐴 ∈ ℕ) |
9 | nn1suc.4 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃)) | |
10 | 9 | sbcieg 3468 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ 𝜃)) |
12 | dfsbcq 3437 | . . . . 5 ⊢ (𝐴 = 1 → ([𝐴 / 𝑥]𝜑 ↔ [1 / 𝑥]𝜑)) | |
13 | 11, 12 | bitr3d 270 | . . . 4 ⊢ (𝐴 = 1 → (𝜃 ↔ [1 / 𝑥]𝜑)) |
14 | 5, 13 | mpbiri 248 | . . 3 ⊢ (𝐴 = 1 → 𝜃) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃)) |
16 | ovex 6678 | . . . . . 6 ⊢ (𝑦 + 1) ∈ V | |
17 | nn1suc.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) | |
18 | 16, 17 | sbcie 3470 | . . . . 5 ⊢ ([(𝑦 + 1) / 𝑥]𝜑 ↔ 𝜒) |
19 | oveq1 6657 | . . . . . 6 ⊢ (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1)) | |
20 | 19 | sbceq1d 3440 | . . . . 5 ⊢ (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
21 | 18, 20 | syl5bbr 274 | . . . 4 ⊢ (𝑦 = (𝐴 − 1) → (𝜒 ↔ [((𝐴 − 1) + 1) / 𝑥]𝜑)) |
22 | nn1suc.6 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝜒) | |
23 | 21, 22 | vtoclga 3272 | . . 3 ⊢ ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑) |
24 | nncn 11028 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
25 | ax-1cn 9994 | . . . . . 6 ⊢ 1 ∈ ℂ | |
26 | npcan 10290 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴) | |
27 | 24, 25, 26 | sylancl 694 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴) |
28 | 27 | sbceq1d 3440 | . . . 4 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
29 | 28, 10 | bitrd 268 | . . 3 ⊢ (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑 ↔ 𝜃)) |
30 | 23, 29 | syl5ib 234 | . 2 ⊢ (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃)) |
31 | nn1m1nn 11040 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) | |
32 | 15, 30, 31 | mpjaod 396 | 1 ⊢ (𝐴 ∈ ℕ → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 [wsbc 3435 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 − cmin 10266 ℕcn 11020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-nn 11021 |
This theorem is referenced by: opsqrlem6 29004 |
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