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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rexzrexnn0 | Structured version Visualization version GIF version | ||
| Description: Rewrite an existential quantification restricted to integers into an existential quantification restricted to naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| rexzrexnn0.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| rexzrexnn0.2 | ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexzrexnn0 | ⊢ (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elznn0 12630 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ ℕ0 ∨ -𝑥 ∈ ℕ0))) | |
| 2 | 1 | simprbi 496 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 ∈ ℕ0 ∨ -𝑥 ∈ ℕ0)) | 
| 3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → (𝑥 ∈ ℕ0 ∨ -𝑥 ∈ ℕ0)) | 
| 4 | simpr 484 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝜑) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0) | |
| 5 | simplr 768 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝜑) ∧ 𝑥 ∈ ℕ0) → 𝜑) | |
| 6 | rexzrexnn0.1 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | equcoms 2018 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | 
| 8 | 7 | bicomd 223 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) | 
| 9 | 8 | rspcev 3621 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝜑) → ∃𝑦 ∈ ℕ0 𝜓) | 
| 10 | 4, 5, 9 | syl2anc 584 | . . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝜑) ∧ 𝑥 ∈ ℕ0) → ∃𝑦 ∈ ℕ0 𝜓) | 
| 11 | 10 | ex 412 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → (𝑥 ∈ ℕ0 → ∃𝑦 ∈ ℕ0 𝜓)) | 
| 12 | simpr 484 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℕ0) → -𝑥 ∈ ℕ0) | |
| 13 | zcn 12620 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
| 14 | 13 | negnegd 11612 | . . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) | 
| 15 | 14 | eqcomd 2742 | . . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 = --𝑥) | 
| 16 | negeq 11501 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
| 17 | 16 | eqeq2d 2747 | . . . . . . . . . . . . 13 ⊢ (𝑦 = -𝑥 → (𝑥 = -𝑦 ↔ 𝑥 = --𝑥)) | 
| 18 | 15, 17 | syl5ibrcom 247 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑦 = -𝑥 → 𝑥 = -𝑦)) | 
| 19 | 18 | imp 406 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) | 
| 20 | rexzrexnn0.2 | . . . . . . . . . . 11 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜒)) | |
| 21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜒)) | 
| 22 | 21 | bicomd 223 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜒 ↔ 𝜑)) | 
| 23 | 22 | adantlr 715 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℕ0) ∧ 𝑦 = -𝑥) → (𝜒 ↔ 𝜑)) | 
| 24 | 12, 23 | rspcedv 3614 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℕ0) → (𝜑 → ∃𝑦 ∈ ℕ0 𝜒)) | 
| 25 | 24 | impancom 451 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → (-𝑥 ∈ ℕ0 → ∃𝑦 ∈ ℕ0 𝜒)) | 
| 26 | 11, 25 | orim12d 966 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → ((𝑥 ∈ ℕ0 ∨ -𝑥 ∈ ℕ0) → (∃𝑦 ∈ ℕ0 𝜓 ∨ ∃𝑦 ∈ ℕ0 𝜒))) | 
| 27 | 3, 26 | mpd 15 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → (∃𝑦 ∈ ℕ0 𝜓 ∨ ∃𝑦 ∈ ℕ0 𝜒)) | 
| 28 | r19.43 3121 | . . . 4 ⊢ (∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒) ↔ (∃𝑦 ∈ ℕ0 𝜓 ∨ ∃𝑦 ∈ ℕ0 𝜒)) | |
| 29 | 27, 28 | sylibr 234 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | 
| 30 | 29 | rexlimiva 3146 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝜑 → ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | 
| 31 | nn0z 12640 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
| 32 | 6 | rspcev 3621 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) | 
| 33 | 31, 32 | sylan 580 | . . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) | 
| 34 | nn0negz 12657 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → -𝑦 ∈ ℤ) | |
| 35 | 20 | rspcev 3621 | . . . . 5 ⊢ ((-𝑦 ∈ ℤ ∧ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) | 
| 36 | 34, 35 | sylan 580 | . . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) | 
| 37 | 33, 36 | jaodan 959 | . . 3 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝜓 ∨ 𝜒)) → ∃𝑥 ∈ ℤ 𝜑) | 
| 38 | 37 | rexlimiva 3146 | . 2 ⊢ (∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) | 
| 39 | 30, 38 | impbii 209 | 1 ⊢ (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ℝcr 11155 -cneg 11494 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 | 
| 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-ltxr 11301 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 | 
| This theorem is referenced by: dvdsrabdioph 42826 | 
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