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Theorem ordpinq 9644
 Description: Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ordpinq ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))

Proof of Theorem ordpinq
StepHypRef Expression
1 brinxp 5104 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴( <pQ ∩ (Q × Q))𝐵))
2 df-ltnq 9619 . . . 4 <Q = ( <pQ ∩ (Q × Q))
32breqi 4589 . . 3 (𝐴 <Q 𝐵𝐴( <pQ ∩ (Q × Q))𝐵)
41, 3syl6bbr 277 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴 <Q 𝐵))
5 relxp 5150 . . . . 5 Rel (N × N)
6 elpqn 9626 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
7 1st2nd 7105 . . . . 5 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
85, 6, 7sylancr 694 . . . 4 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
9 elpqn 9626 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
10 1st2nd 7105 . . . . 5 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
115, 9, 10sylancr 694 . . . 4 (𝐵Q𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
128, 11breqan12d 4599 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
13 ordpipq 9643 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
1412, 13syl6bb 275 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
154, 14bitr3d 269 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ⟨cop 4131   class class class wbr 4583   × cxp 5036  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   ·N cmi 9547
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