Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11289 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
2 | elznn0nn 11268 | . . . . . . 7 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | |
3 | 1, 2 | sylib 207 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) |
4 | simpr 476 | . . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ) → --𝑦 ∈ ℕ) | |
5 | 4 | orim2i 539 | . . . . . 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 11259 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
8 | 7 | negnegd 10262 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → --𝑦 = 𝑦) |
9 | 8 | eleq1d 2672 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (--𝑦 ∈ ℕ ↔ 𝑦 ∈ ℕ)) |
10 | 9 | orbi2d 734 | . . . . 5 ⊢ (𝑦 ∈ ℤ → ((-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ) ↔ (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ))) |
11 | 6, 10 | mpbid 221 | . . . 4 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ)) |
12 | zindd.1 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
13 | 12 | imbi2d 329 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜓))) |
14 | zindd.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
15 | 14 | imbi2d 329 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜒))) |
16 | zindd.3 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) | |
17 | 16 | imbi2d 329 | . . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜏))) |
18 | zindd.4 | . . . . . . . 8 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) | |
19 | 18 | imbi2d 329 | . . . . . . 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 11348 | . . . . . 6 ⊢ (-𝑦 ∈ ℕ0 → (𝜁 → 𝜃)) |
25 | 24 | com12 32 | . . . . 5 ⊢ (𝜁 → (-𝑦 ∈ ℕ0 → 𝜃)) |
26 | nnnn0 11176 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
27 | 13, 15, 17, 15, 20, 23 | nn0ind 11348 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → 𝜒)) |
28 | 26, 27 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝜁 → 𝜒)) |
29 | 28 | com12 32 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜒)) |
30 | zindd.8 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) | |
31 | 29, 30 | mpdd 42 | . . . . 5 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜃)) |
32 | 25, 31 | jaod 394 | . . . 4 ⊢ (𝜁 → ((-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ) → 𝜃)) |
33 | 11, 32 | syl5 33 | . . 3 ⊢ (𝜁 → (𝑦 ∈ ℤ → 𝜃)) |
34 | 33 | ralrimiv 2948 | . 2 ⊢ (𝜁 → ∀𝑦 ∈ ℤ 𝜃) |
35 | znegcl 11289 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
36 | negeq 10152 | . . . . . . . . 9 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
37 | zcn 11259 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
38 | 37 | negnegd 10262 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
39 | 36, 38 | sylan9eqr 2666 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → -𝑦 = 𝑥) |
40 | 39 | eqcomd 2616 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) |
41 | 40, 18 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜃)) |
42 | 41 | bicomd 212 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜃 ↔ 𝜑)) |
43 | 35, 42 | rspcdv 3285 | . . . 4 ⊢ (𝑥 ∈ ℤ → (∀𝑦 ∈ ℤ 𝜃 → 𝜑)) |
44 | 43 | com12 32 | . . 3 ⊢ (∀𝑦 ∈ ℤ 𝜃 → (𝑥 ∈ ℤ → 𝜑)) |
45 | 44 | ralrimiv 2948 | . 2 ⊢ (∀𝑦 ∈ ℤ 𝜃 → ∀𝑥 ∈ ℤ 𝜑) |
46 | zindd.5 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
47 | 46 | rspccv 3279 | . 2 ⊢ (∀𝑥 ∈ ℤ 𝜑 → (𝐴 ∈ ℤ → 𝜂)) |
48 | 34, 45, 47 | 3syl 18 | 1 ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 -cneg 10146 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: efexp 14670 pcexp 15402 mulgaddcom 17387 mulginvcom 17388 mulgneg2 17398 mulgass2 18424 cnfldmulg 19597 clmmulg 22709 xrsmulgzz 29009 |
Copyright terms: Public domain | W3C validator |