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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 12337 | . . 3 ⊢ ℚ = ( / “ (ℤ × ℕ)) | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝐴 ∈ ℚ ↔ 𝐴 ∈ ( / “ (ℤ × ℕ))) |
3 | df-div 11286 | . . . 4 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
4 | riotaex 7107 | . . . 4 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
5 | 3, 4 | fnmpoi 7757 | . . 3 ⊢ / Fn (ℂ × (ℂ ∖ {0})) |
6 | zsscn 11977 | . . . 4 ⊢ ℤ ⊆ ℂ | |
7 | nncn 11634 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
8 | nnne0 11659 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0) | |
9 | eldifsn 4711 | . . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
10 | 7, 8, 9 | sylanbrc 583 | . . . . 5 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0})) |
11 | 10 | ssriv 3968 | . . . 4 ⊢ ℕ ⊆ (ℂ ∖ {0}) |
12 | xpss12 5563 | . . . 4 ⊢ ((ℤ ⊆ ℂ ∧ ℕ ⊆ (ℂ ∖ {0})) → (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0}))) | |
13 | 6, 11, 12 | mp2an 688 | . . 3 ⊢ (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0})) |
14 | ovelimab 7315 | . . 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 688 | . 2 ⊢ (𝐴 ∈ ( / “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
16 | 2, 15 | bitri 276 | 1 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 × cxp 5546 “ cima 5551 Fn wfn 6343 ℩crio 7102 (class class class)co 7145 ℂcc 10523 0cc0 10525 · cmul 10530 / cdiv 11285 ℕcn 11626 ℤcz 11969 ℚcq 12336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-div 11286 df-nn 11627 df-z 11970 df-q 12337 |
This theorem is referenced by: qmulz 12339 znq 12340 qre 12341 qexALT 12351 qaddcl 12352 qnegcl 12353 qmulcl 12354 qreccl 12356 elpq 12362 eirr 15546 qnnen 15554 sqrt2irr 15590 qredeu 15990 pceu 16171 pcqmul 16178 pcqcl 16181 pcneg 16198 pcz 16205 pcadd 16213 qsssubdrg 20532 ostthlem1 26130 ipasslem5 28539 |
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