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Mirrors > Home > ILE Home > Th. List > enqbreq | GIF version |
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
enqbreq | ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q 〈𝐶, 𝐷〉 ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 6537 | . 2 ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | |
2 | 1 | ecopoveq 6224 | 1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q 〈𝐶, 𝐷〉 ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 (class class class)co 5532 Ncnpi 6462 ·N cmi 6464 ~Q ceq 6469 |
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-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-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 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 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-ral 2353 df-rex 2354 df-v 2603 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-br 3786 df-opab 3840 df-xp 4369 df-iota 4887 df-fv 4930 df-ov 5535 df-enq 6537 |
This theorem is referenced by: enqbreq2 6547 enqeceq 6549 enqdc 6551 addcmpblnq 6557 mulcmpblnq 6558 mulcanenq 6575 |
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