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Mirrors > Home > MPE Home > Th. List > elq | Structured version Visualization version GIF version |
Description: Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.) |
Ref | Expression |
---|---|
elq | ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-q 12096 | . . 3 ⊢ ℚ = ( / “ (ℤ × ℕ)) | |
2 | 1 | eleq2i 2851 | . 2 ⊢ (𝐴 ∈ ℚ ↔ 𝐴 ∈ ( / “ (ℤ × ℕ))) |
3 | df-div 11033 | . . . 4 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
4 | riotaex 6887 | . . . 4 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
5 | 3, 4 | fnmpt2i 7519 | . . 3 ⊢ / Fn (ℂ × (ℂ ∖ {0})) |
6 | zsscn 11736 | . . . 4 ⊢ ℤ ⊆ ℂ | |
7 | nncn 11383 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
8 | nnne0 11410 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0) | |
9 | eldifsn 4550 | . . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
10 | 7, 8, 9 | sylanbrc 578 | . . . . 5 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0})) |
11 | 10 | ssriv 3825 | . . . 4 ⊢ ℕ ⊆ (ℂ ∖ {0}) |
12 | xpss12 5370 | . . . 4 ⊢ ((ℤ ⊆ ℂ ∧ ℕ ⊆ (ℂ ∖ {0})) → (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0}))) | |
13 | 6, 11, 12 | mp2an 682 | . . 3 ⊢ (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0})) |
14 | ovelimab 7089 | . . 3 ⊢ (( / Fn (ℂ × (ℂ ∖ {0})) ∧ (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0}))) → (𝐴 ∈ ( / “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))) | |
15 | 5, 13, 14 | mp2an 682 | . 2 ⊢ (𝐴 ∈ ( / “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
16 | 2, 15 | bitri 267 | 1 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 ∖ cdif 3789 ⊆ wss 3792 {csn 4398 × cxp 5353 “ cima 5358 Fn wfn 6130 ℩crio 6882 (class class class)co 6922 ℂcc 10270 0cc0 10272 · cmul 10277 / cdiv 11032 ℕcn 11374 ℤcz 11728 ℚcq 12095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-neg 10609 df-div 11033 df-nn 11375 df-z 11729 df-q 12096 |
This theorem is referenced by: qmulz 12098 znq 12099 qre 12100 zqOLD 12102 qexALT 12111 qaddcl 12112 qnegcl 12113 qmulcl 12114 qreccl 12116 elpq 12122 eirr 15337 qnnen 15346 sqrt2irr 15382 qredeu 15777 pceu 15955 pcqmul 15962 pcqcl 15965 pcneg 15982 pcz 15989 pcadd 15997 qsssubdrg 20201 ostthlem1 25768 ipasslem5 28262 |
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