Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnindf | Structured version Visualization version GIF version |
Description: Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.) |
Ref | Expression |
---|---|
nnindf.x | ⊢ Ⅎ𝑦𝜑 |
nnindf.1 | ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
nnindf.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
nnindf.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
nnindf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
nnindf.5 | ⊢ 𝜓 |
nnindf.6 | ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
nnindf | ⊢ (𝐴 ∈ ℕ → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11031 | . . . . . 6 ⊢ 1 ∈ ℕ | |
2 | nnindf.5 | . . . . . 6 ⊢ 𝜓 | |
3 | nnindf.1 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) | |
4 | 3 | elrab 3363 | . . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓)) |
5 | 1, 2, 4 | mpbir2an 955 | . . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
6 | elrabi 3359 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ) | |
7 | peano2nn 11032 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ) | |
8 | 7 | a1d 25 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)) |
9 | nnindf.6 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃)) | |
10 | 8, 9 | anim12d 586 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃))) |
11 | nnindf.2 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
12 | 11 | elrab 3363 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒)) |
13 | nnindf.3 | . . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) | |
14 | 13 | elrab 3363 | . . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃)) |
15 | 10, 12, 14 | 3imtr4g 285 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
16 | 6, 15 | mpcom 38 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
17 | 16 | rgen 2922 | . . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
18 | nnindf.x | . . . . . . . 8 ⊢ Ⅎ𝑦𝜑 | |
19 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑦ℕ | |
20 | 18, 19 | nfrab 3123 | . . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑} |
21 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑} | |
22 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} | |
23 | 20 | nfel2 2781 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
24 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) | |
25 | 24 | eleq1d 2686 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})) |
26 | 20, 21, 22, 23, 25 | cbvralf 3165 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
27 | 17, 26 | mpbi 220 | . . . . 5 ⊢ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} |
28 | peano5nni 11023 | . . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}) | |
29 | 5, 27, 28 | mp2an 708 | . . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑} |
30 | 29 | sseli 3599 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑}) |
31 | nnindf.4 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
32 | 31 | elrab 3363 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏)) |
33 | 30, 32 | sylib 208 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏)) |
34 | 33 | simprd 479 | 1 ⊢ (𝐴 ∈ ℕ → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 (class class class)co 6650 1c1 9937 + caddc 9939 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-1cn 9994 |
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-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-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 |
This theorem is referenced by: nn0min 29567 |
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