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Theorem ordpipqqs 7194
 Description: Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
Assertion
Ref Expression
ordpipqqs (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q <Q [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶)))

Proof of Theorem ordpipqqs
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enqex 7180 . 2 ~Q ∈ V
2 enqer 7178 . 2 ~Q Er (N × N)
3 df-nqqs 7168 . 2 Q = ((N × N) / ~Q )
4 df-ltnqqs 7173 . 2 <Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}
5 enqeceq 7179 . . . . 5 (((𝑧N𝑤N) ∧ (𝐴N𝐵N)) → ([⟨𝑧, 𝑤⟩] ~Q = [⟨𝐴, 𝐵⟩] ~Q ↔ (𝑧 ·N 𝐵) = (𝑤 ·N 𝐴)))
6 enqeceq 7179 . . . . . 6 (((𝑣N𝑢N) ∧ (𝐶N𝐷N)) → ([⟨𝑣, 𝑢⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (𝑣 ·N 𝐷) = (𝑢 ·N 𝐶)))
7 eqcom 2141 . . . . . 6 ((𝑣 ·N 𝐷) = (𝑢 ·N 𝐶) ↔ (𝑢 ·N 𝐶) = (𝑣 ·N 𝐷))
86, 7syl6bb 195 . . . . 5 (((𝑣N𝑢N) ∧ (𝐶N𝐷N)) → ([⟨𝑣, 𝑢⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (𝑢 ·N 𝐶) = (𝑣 ·N 𝐷)))
95, 8bi2anan9 595 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (([⟨𝑧, 𝑤⟩] ~Q = [⟨𝐴, 𝐵⟩] ~Q ∧ [⟨𝑣, 𝑢⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ) ↔ ((𝑧 ·N 𝐵) = (𝑤 ·N 𝐴) ∧ (𝑢 ·N 𝐶) = (𝑣 ·N 𝐷))))
10 oveq12 5783 . . . . 5 (((𝑧 ·N 𝐵) = (𝑤 ·N 𝐴) ∧ (𝑢 ·N 𝐶) = (𝑣 ·N 𝐷)) → ((𝑧 ·N 𝐵) ·N (𝑢 ·N 𝐶)) = ((𝑤 ·N 𝐴) ·N (𝑣 ·N 𝐷)))
11 simplll 522 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝑧N)
12 simprlr 527 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝑢N)
13 simplrr 525 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝐵N)
14 mulcompig 7151 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥))
1514adantl 275 . . . . . . 7 (((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) ∧ (𝑥N𝑦N)) → (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥))
16 mulasspig 7152 . . . . . . . 8 ((𝑥N𝑦N𝑓N) → ((𝑥 ·N 𝑦) ·N 𝑓) = (𝑥 ·N (𝑦 ·N 𝑓)))
1716adantl 275 . . . . . . 7 (((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) ∧ (𝑥N𝑦N𝑓N)) → ((𝑥 ·N 𝑦) ·N 𝑓) = (𝑥 ·N (𝑦 ·N 𝑓)))
18 simprrl 528 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝐶N)
19 mulclpi 7148 . . . . . . . 8 ((𝑥N𝑦N) → (𝑥 ·N 𝑦) ∈ N)
2019adantl 275 . . . . . . 7 (((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) ∧ (𝑥N𝑦N)) → (𝑥 ·N 𝑦) ∈ N)
2111, 12, 13, 15, 17, 18, 20caov4d 5955 . . . . . 6 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → ((𝑧 ·N 𝑢) ·N (𝐵 ·N 𝐶)) = ((𝑧 ·N 𝐵) ·N (𝑢 ·N 𝐶)))
22 simpllr 523 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝑤N)
23 simprll 526 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝑣N)
24 simplrl 524 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝐴N)
25 simprrr 529 . . . . . . 7 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → 𝐷N)
2622, 23, 24, 15, 17, 25, 20caov4d 5955 . . . . . 6 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → ((𝑤 ·N 𝑣) ·N (𝐴 ·N 𝐷)) = ((𝑤 ·N 𝐴) ·N (𝑣 ·N 𝐷)))
2721, 26eqeq12d 2154 . . . . 5 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (((𝑧 ·N 𝑢) ·N (𝐵 ·N 𝐶)) = ((𝑤 ·N 𝑣) ·N (𝐴 ·N 𝐷)) ↔ ((𝑧 ·N 𝐵) ·N (𝑢 ·N 𝐶)) = ((𝑤 ·N 𝐴) ·N (𝑣 ·N 𝐷))))
2810, 27syl5ibr 155 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (((𝑧 ·N 𝐵) = (𝑤 ·N 𝐴) ∧ (𝑢 ·N 𝐶) = (𝑣 ·N 𝐷)) → ((𝑧 ·N 𝑢) ·N (𝐵 ·N 𝐶)) = ((𝑤 ·N 𝑣) ·N (𝐴 ·N 𝐷))))
299, 28sylbid 149 . . 3 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (([⟨𝑧, 𝑤⟩] ~Q = [⟨𝐴, 𝐵⟩] ~Q ∧ [⟨𝑣, 𝑢⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ) → ((𝑧 ·N 𝑢) ·N (𝐵 ·N 𝐶)) = ((𝑤 ·N 𝑣) ·N (𝐴 ·N 𝐷))))
30 ltmpig 7159 . . . . 5 ((𝑥N𝑦N𝑓N) → (𝑥 <N 𝑦 ↔ (𝑓 ·N 𝑥) <N (𝑓 ·N 𝑦)))
3130adantl 275 . . . 4 (((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) ∧ (𝑥N𝑦N𝑓N)) → (𝑥 <N 𝑦 ↔ (𝑓 ·N 𝑥) <N (𝑓 ·N 𝑦)))
3220, 11, 12caovcld 5924 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (𝑧 ·N 𝑢) ∈ N)
3320, 13, 18caovcld 5924 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (𝐵 ·N 𝐶) ∈ N)
3420, 22, 23caovcld 5924 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (𝑤 ·N 𝑣) ∈ N)
3520, 24, 25caovcld 5924 . . . 4 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (𝐴 ·N 𝐷) ∈ N)
3631, 32, 33, 34, 15, 35caovord3d 5941 . . 3 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (((𝑧 ·N 𝑢) ·N (𝐵 ·N 𝐶)) = ((𝑤 ·N 𝑣) ·N (𝐴 ·N 𝐷)) → ((𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣) ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶))))
3729, 36syld 45 . 2 ((((𝑧N𝑤N) ∧ (𝐴N𝐵N)) ∧ ((𝑣N𝑢N) ∧ (𝐶N𝐷N))) → (([⟨𝑧, 𝑤⟩] ~Q = [⟨𝐴, 𝐵⟩] ~Q ∧ [⟨𝑣, 𝑢⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ) → ((𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣) ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶))))
381, 2, 3, 4, 37brecop 6519 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q <Q [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929  (class class class)co 5774  [cec 6427  Ncnpi 7092   ·N cmi 7094
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