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Mirrors > Home > MPE Home > Th. List > zindd | Structured version Visualization version GIF version |
Description: Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
zindd.1 | ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) |
zindd.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
zindd.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) |
zindd.4 | ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) |
zindd.5 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) |
zindd.6 | ⊢ (𝜁 → 𝜓) |
zindd.7 | ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) |
zindd.8 | ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) |
Ref | Expression |
---|---|
zindd | ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl 12006 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
2 | elznn0nn 11984 | . . . . . . 7 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | |
3 | 1, 2 | sylib 219 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) |
4 | simpr 485 | . . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ) → --𝑦 ∈ ℕ) | |
5 | 4 | orim2i 904 | . . . . . 6 ⊢ ((-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ)) → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
7 | zcn 11975 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
8 | 7 | negnegd 10977 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → --𝑦 = 𝑦) |
9 | 8 | eleq1d 2897 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (--𝑦 ∈ ℕ ↔ 𝑦 ∈ ℕ)) |
10 | 9 | orbi2d 909 | . . . . 5 ⊢ (𝑦 ∈ ℤ → ((-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ) ↔ (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ))) |
11 | 6, 10 | mpbid 233 | . . . 4 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ)) |
12 | zindd.1 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
13 | 12 | imbi2d 342 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜓))) |
14 | zindd.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
15 | 14 | imbi2d 342 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜒))) |
16 | zindd.3 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) | |
17 | 16 | imbi2d 342 | . . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜏))) |
18 | zindd.4 | . . . . . . . 8 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) | |
19 | 18 | imbi2d 342 | . . . . . . 7 ⊢ (𝑥 = -𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜃))) |
20 | zindd.6 | . . . . . . 7 ⊢ (𝜁 → 𝜓) | |
21 | zindd.7 | . . . . . . . . 9 ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) | |
22 | 21 | com12 32 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → (𝜒 → 𝜏))) |
23 | 22 | a2d 29 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → ((𝜁 → 𝜒) → (𝜁 → 𝜏))) |
24 | 13, 15, 17, 19, 20, 23 | nn0ind 12066 | . . . . . 6 ⊢ (-𝑦 ∈ ℕ0 → (𝜁 → 𝜃)) |
25 | 24 | com12 32 | . . . . 5 ⊢ (𝜁 → (-𝑦 ∈ ℕ0 → 𝜃)) |
26 | 13, 15, 17, 15, 20, 23 | nn0ind 12066 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → 𝜒)) |
27 | nnnn0 11893 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
28 | 26, 27 | syl11 33 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜒)) |
29 | zindd.8 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) | |
30 | 28, 29 | mpdd 43 | . . . . 5 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜃)) |
31 | 25, 30 | jaod 853 | . . . 4 ⊢ (𝜁 → ((-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ) → 𝜃)) |
32 | 11, 31 | syl5 34 | . . 3 ⊢ (𝜁 → (𝑦 ∈ ℤ → 𝜃)) |
33 | 32 | ralrimiv 3181 | . 2 ⊢ (𝜁 → ∀𝑦 ∈ ℤ 𝜃) |
34 | znegcl 12006 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
35 | negeq 10867 | . . . . . . . . 9 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
36 | zcn 11975 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
37 | 36 | negnegd 10977 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
38 | 35, 37 | sylan9eqr 2878 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → -𝑦 = 𝑥) |
39 | 38 | eqcomd 2827 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) |
40 | 39, 18 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜃)) |
41 | 40 | bicomd 224 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜃 ↔ 𝜑)) |
42 | 34, 41 | rspcdv 3614 | . . . 4 ⊢ (𝑥 ∈ ℤ → (∀𝑦 ∈ ℤ 𝜃 → 𝜑)) |
43 | 42 | com12 32 | . . 3 ⊢ (∀𝑦 ∈ ℤ 𝜃 → (𝑥 ∈ ℤ → 𝜑)) |
44 | 43 | ralrimiv 3181 | . 2 ⊢ (∀𝑦 ∈ ℤ 𝜃 → ∀𝑥 ∈ ℤ 𝜑) |
45 | zindd.5 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
46 | 45 | rspccv 3619 | . 2 ⊢ (∀𝑥 ∈ ℤ 𝜑 → (𝐴 ∈ ℤ → 𝜂)) |
47 | 33, 44, 46 | 3syl 18 | 1 ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∀wral 3138 (class class class)co 7145 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 -cneg 10860 ℕcn 11627 ℕ0cn0 11886 ℤcz 11970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 |
This theorem is referenced by: efexp 15444 pcexp 16186 mulgaddcom 18191 mulginvcom 18192 mulgneg2 18201 mulgass2 19282 cnfldmulg 20507 clmmulg 23634 xrsmulgzz 30593 |
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