Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elq | 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 8705 | . . . 4 ⊢ ℚ = ( / “ (ℤ × ℕ)) | |
2 | 1 | eleq2i 2145 | . . 3 ⊢ (𝐴 ∈ ℚ ↔ 𝐴 ∈ ( / “ (ℤ × ℕ))) |
3 | resima 4661 | . . . 4 ⊢ (( / ↾ (ℤ × ℕ)) “ (ℤ × ℕ)) = ( / “ (ℤ × ℕ)) | |
4 | 3 | eleq2i 2145 | . . 3 ⊢ (𝐴 ∈ (( / ↾ (ℤ × ℕ)) “ (ℤ × ℕ)) ↔ 𝐴 ∈ ( / “ (ℤ × ℕ))) |
5 | divfnzn 8706 | . . . 4 ⊢ ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ) | |
6 | ssid 3018 | . . . 4 ⊢ (ℤ × ℕ) ⊆ (ℤ × ℕ) | |
7 | ovelimab 5671 | . . . 4 ⊢ ((( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ) ∧ (ℤ × ℕ) ⊆ (ℤ × ℕ)) → (𝐴 ∈ (( / ↾ (ℤ × ℕ)) “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥( / ↾ (ℤ × ℕ))𝑦))) | |
8 | 5, 6, 7 | mp2an 416 | . . 3 ⊢ (𝐴 ∈ (( / ↾ (ℤ × ℕ)) “ (ℤ × ℕ)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥( / ↾ (ℤ × ℕ))𝑦)) |
9 | 2, 4, 8 | 3bitr2i 206 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥( / ↾ (ℤ × ℕ))𝑦)) |
10 | ovres 5660 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥( / ↾ (ℤ × ℕ))𝑦) = (𝑥 / 𝑦)) | |
11 | 10 | eqeq2d 2092 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥( / ↾ (ℤ × ℕ))𝑦) ↔ 𝐴 = (𝑥 / 𝑦))) |
12 | 11 | 2rexbiia 2382 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥( / ↾ (ℤ × ℕ))𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
13 | 9, 12 | bitri 182 | 1 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 ⊆ wss 2973 × cxp 4361 ↾ cres 4365 “ cima 4366 Fn wfn 4917 (class class class)co 5532 / cdiv 7760 ℕcn 8039 ℤcz 8351 ℚcq 8704 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-z 8352 df-q 8705 |
This theorem is referenced by: qmulz 8708 znq 8709 qre 8710 zq 8711 qaddcl 8720 qnegcl 8721 qmulcl 8722 qapne 8724 qreccl 8727 qtri3or 9252 qredeu 10479 sqrt2irr 10541 sqrt2irrap 10558 |
Copyright terms: Public domain | W3C validator |