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Mathbox for Stefan O'Rear |
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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 12626 | . . . . . . 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 769 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝜑) ∧ 𝑥 ∈ ℕ0) → 𝜑) | |
6 | rexzrexnn0.1 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | 6 | equcoms 2017 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
8 | 7 | bicomd 223 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
9 | 8 | rspcev 3622 | . . . . . . . 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 12616 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
14 | 13 | negnegd 11609 | . . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
15 | 14 | eqcomd 2741 | . . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 = --𝑥) |
16 | negeq 11498 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
17 | 16 | eqeq2d 2746 | . . . . . . . . . . . . 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 3615 | . . . . . . 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 3120 | . . . 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 3145 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝜑 → ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) |
31 | nn0z 12636 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
32 | 6 | rspcev 3622 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) |
33 | 31, 32 | sylan 580 | . . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) |
34 | nn0negz 12653 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → -𝑦 ∈ ℤ) | |
35 | 20 | rspcev 3622 | . . . . 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 3145 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 ℝcr 11152 -cneg 11491 ℕ0cn0 12524 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 |
This theorem is referenced by: dvdsrabdioph 42798 |
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