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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 12577 | . . . . . . 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 766 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝜑) ∧ 𝑥 ∈ ℕ0) → 𝜑) | |
6 | rexzrexnn0.1 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | 6 | equcoms 2015 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
8 | 7 | bicomd 222 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
9 | 8 | rspcev 3606 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝜑) → ∃𝑦 ∈ ℕ0 𝜓) |
10 | 4, 5, 9 | syl2anc 583 | . . . . . . 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 12567 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
14 | 13 | negnegd 11566 | . . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
15 | 14 | eqcomd 2732 | . . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → 𝑥 = --𝑥) |
16 | negeq 11456 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
17 | 16 | eqeq2d 2737 | . . . . . . . . . . . . 13 ⊢ (𝑦 = -𝑥 → (𝑥 = -𝑦 ↔ 𝑥 = --𝑥)) |
18 | 15, 17 | syl5ibrcom 246 | . . . . . . . . . . . 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 222 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜒 ↔ 𝜑)) |
23 | 22 | adantlr 712 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℕ0) ∧ 𝑦 = -𝑥) → (𝜒 ↔ 𝜑)) |
24 | 12, 23 | rspcedv 3599 | . . . . . . 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 961 | . . . . 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 3116 | . . . 4 ⊢ (∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒) ↔ (∃𝑦 ∈ ℕ0 𝜓 ∨ ∃𝑦 ∈ ℕ0 𝜒)) | |
29 | 27, 28 | sylibr 233 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝜑) → ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) |
30 | 29 | rexlimiva 3141 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝜑 → ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) |
31 | nn0z 12587 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
32 | 6 | rspcev 3606 | . . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) |
33 | 31, 32 | sylan 579 | . . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝜓) → ∃𝑥 ∈ ℤ 𝜑) |
34 | nn0negz 12604 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → -𝑦 ∈ ℤ) | |
35 | 20 | rspcev 3606 | . . . . 5 ⊢ ((-𝑦 ∈ ℤ ∧ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) |
36 | 34, 35 | sylan 579 | . . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) |
37 | 33, 36 | jaodan 954 | . . 3 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝜓 ∨ 𝜒)) → ∃𝑥 ∈ ℤ 𝜑) |
38 | 37 | rexlimiva 3141 | . 2 ⊢ (∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒) → ∃𝑥 ∈ ℤ 𝜑) |
39 | 30, 38 | impbii 208 | 1 ⊢ (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ℝcr 11111 -cneg 11449 ℕ0cn0 12476 ℤcz 12562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 |
This theorem is referenced by: dvdsrabdioph 42131 |
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