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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 12654 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
| 2 | elznn0nn 12629 | . . . . . . 7 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | |
| 3 | 1, 2 | sylib 218 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | 
| 4 | simpr 484 | . . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ) → --𝑦 ∈ ℕ) | |
| 5 | 4 | orim2i 910 | . . . . . 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 12620 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
| 8 | 7 | negnegd 11612 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → --𝑦 = 𝑦) | 
| 9 | 8 | eleq1d 2825 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (--𝑦 ∈ ℕ ↔ 𝑦 ∈ ℕ)) | 
| 10 | 9 | orbi2d 915 | . . . . 5 ⊢ (𝑦 ∈ ℤ → ((-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ) ↔ (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ))) | 
| 11 | 6, 10 | mpbid 232 | . . . 4 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ)) | 
| 12 | zindd.1 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
| 13 | 12 | imbi2d 340 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜓))) | 
| 14 | zindd.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
| 15 | 14 | imbi2d 340 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜒))) | 
| 16 | zindd.3 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) | |
| 17 | 16 | imbi2d 340 | . . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜏))) | 
| 18 | zindd.4 | . . . . . . . 8 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) | |
| 19 | 18 | imbi2d 340 | . . . . . . 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 12715 | . . . . . 6 ⊢ (-𝑦 ∈ ℕ0 → (𝜁 → 𝜃)) | 
| 25 | 24 | com12 32 | . . . . 5 ⊢ (𝜁 → (-𝑦 ∈ ℕ0 → 𝜃)) | 
| 26 | 13, 15, 17, 15, 20, 23 | nn0ind 12715 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → 𝜒)) | 
| 27 | nnnn0 12535 | . . . . . . 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 859 | . . . 4 ⊢ (𝜁 → ((-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ) → 𝜃)) | 
| 32 | 11, 31 | syl5 34 | . . 3 ⊢ (𝜁 → (𝑦 ∈ ℤ → 𝜃)) | 
| 33 | 32 | ralrimiv 3144 | . 2 ⊢ (𝜁 → ∀𝑦 ∈ ℤ 𝜃) | 
| 34 | znegcl 12654 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
| 35 | negeq 11501 | . . . . . . . . 9 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
| 36 | zcn 12620 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
| 37 | 36 | negnegd 11612 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) | 
| 38 | 35, 37 | sylan9eqr 2798 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → -𝑦 = 𝑥) | 
| 39 | 38 | eqcomd 2742 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) | 
| 40 | 39, 18 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜃)) | 
| 41 | 40 | bicomd 223 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜃 ↔ 𝜑)) | 
| 42 | 34, 41 | rspcdv 3613 | . . . 4 ⊢ (𝑥 ∈ ℤ → (∀𝑦 ∈ ℤ 𝜃 → 𝜑)) | 
| 43 | 42 | com12 32 | . . 3 ⊢ (∀𝑦 ∈ ℤ 𝜃 → (𝑥 ∈ ℤ → 𝜑)) | 
| 44 | 43 | ralrimiv 3144 | . 2 ⊢ (∀𝑦 ∈ ℤ 𝜃 → ∀𝑥 ∈ ℤ 𝜑) | 
| 45 | zindd.5 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
| 46 | 45 | rspccv 3618 | . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∀wral 3060 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 -cneg 11494 ℕcn 12267 ℕ0cn0 12528 ℤcz 12615 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 | 
| This theorem is referenced by: efexp 16138 pcexp 16898 mulgaddcom 19117 mulginvcom 19118 mulgneg2 19127 mulgass2 20307 cnfldmulg 21417 clmmulg 25135 xrsmulgzz 33012 | 
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