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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 12829 | . . 3 ⊢ ℚ = ( / “ (ℤ × ℕ)) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ ℚ ↔ 𝐴 ∈ ( / “ (ℤ × ℕ))) |
3 | df-div 11772 | . . . 4 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
4 | riotaex 7312 | . . . 4 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
5 | 3, 4 | fnmpoi 7995 | . . 3 ⊢ / Fn (ℂ × (ℂ ∖ {0})) |
6 | zsscn 12466 | . . . 4 ⊢ ℤ ⊆ ℂ | |
7 | nncn 12120 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
8 | nnne0 12146 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0) | |
9 | eldifsn 4746 | . . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
10 | 7, 8, 9 | sylanbrc 584 | . . . . 5 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℂ ∖ {0})) |
11 | 10 | ssriv 3947 | . . . 4 ⊢ ℕ ⊆ (ℂ ∖ {0}) |
12 | xpss12 5647 | . . . 4 ⊢ ((ℤ ⊆ ℂ ∧ ℕ ⊆ (ℂ ∖ {0})) → (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0}))) | |
13 | 6, 11, 12 | mp2an 691 | . . 3 ⊢ (ℤ × ℕ) ⊆ (ℂ × (ℂ ∖ {0})) |
14 | ovelimab 7527 | . . 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 691 | . 2 ⊢ (𝐴 ∈ ( / “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
16 | 2, 15 | bitri 275 | 1 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∃wrex 3072 ∖ cdif 3906 ⊆ wss 3909 {csn 4585 × cxp 5630 “ cima 5635 Fn wfn 6489 ℩crio 7307 (class class class)co 7352 ℂcc 11008 0cc0 11010 · cmul 11015 / cdiv 11771 ℕcn 12112 ℤcz 12458 ℚcq 12828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-neg 11347 df-div 11772 df-nn 12113 df-z 12459 df-q 12829 |
This theorem is referenced by: qmulz 12831 znq 12832 qre 12833 qexALT 12844 qaddcl 12845 qnegcl 12846 qmulcl 12847 qreccl 12849 elpq 12855 eirr 16047 qnnen 16055 sqrt2irr 16091 qredeu 16494 pceu 16678 pcqmul 16685 pcqcl 16688 pcneg 16706 pcz 16713 pcadd 16721 qsssubdrg 20809 ostthlem1 26927 ipasslem5 29606 1fldgenq 31913 |
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