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Mirrors > Home > MPE Home > Th. List > enqbreq | Structured version Visualization version GIF version |
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
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 10321 | . 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 8387 | 1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q 〈𝐶, 𝐷〉 ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 (class class class)co 7145 Ncnpi 10254 ·N cmi 10256 ~Q ceq 10261 |
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-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-iota 6307 df-fv 6356 df-ov 7148 df-enq 10321 |
This theorem is referenced by: enqbreq2 10330 nqereu 10339 adderpqlem 10364 mulerpqlem 10365 mulcanenq 10370 |
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