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Theorem ordpipqqs 6358
 Description: Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
Assertion
Ref Expression
ordpipqqs (((A N B N) (𝐶 N 𝐷 N)) → ([⟨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 6344 . 2 ~Q V
2 enqer 6342 . 2 ~Q Er (N × N)
3 df-nqqs 6332 . 2 Q = ((N × N) / ~Q )
4 df-ltnqqs 6337 . 2 <Q = {⟨x, y⟩ ∣ ((x Q y Q) zwvu((x = [⟨z, w⟩] ~Q y = [⟨v, u⟩] ~Q ) (z ·N u) <N (w ·N v)))}
5 enqeceq 6343 . . . . 5 (((z N w N) (A N B N)) → ([⟨z, w⟩] ~Q = [⟨A, B⟩] ~Q ↔ (z ·N B) = (w ·N A)))
6 enqeceq 6343 . . . . . 6 (((v N u N) (𝐶 N 𝐷 N)) → ([⟨v, u⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (v ·N 𝐷) = (u ·N 𝐶)))
7 eqcom 2039 . . . . . 6 ((v ·N 𝐷) = (u ·N 𝐶) ↔ (u ·N 𝐶) = (v ·N 𝐷))
86, 7syl6bb 185 . . . . 5 (((v N u N) (𝐶 N 𝐷 N)) → ([⟨v, u⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (u ·N 𝐶) = (v ·N 𝐷)))
95, 8bi2anan9 538 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (([⟨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 5464 . . . . 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 485 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → z N)
12 simprlr 490 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → u N)
13 simplrr 488 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → B N)
14 mulcompig 6315 . . . . . . . 8 ((x N y N) → (x ·N y) = (y ·N x))
1514adantl 262 . . . . . . 7 (((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) (x N y N)) → (x ·N y) = (y ·N x))
16 mulasspig 6316 . . . . . . . 8 ((x N y N f N) → ((x ·N y) ·N f) = (x ·N (y ·N f)))
1716adantl 262 . . . . . . 7 (((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) (x N y N f N)) → ((x ·N y) ·N f) = (x ·N (y ·N f)))
18 simprrl 491 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → 𝐶 N)
19 mulclpi 6312 . . . . . . . 8 ((x N y N) → (x ·N y) N)
2019adantl 262 . . . . . . 7 (((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) (x N y N)) → (x ·N y) N)
2111, 12, 13, 15, 17, 18, 20caov4d 5627 . . . . . 6 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → ((z ·N u) ·N (B ·N 𝐶)) = ((z ·N B) ·N (u ·N 𝐶)))
22 simpllr 486 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → w N)
23 simprll 489 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → v N)
24 simplrl 487 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → A N)
25 simprrr 492 . . . . . . 7 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → 𝐷 N)
2622, 23, 24, 15, 17, 25, 20caov4d 5627 . . . . . 6 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → ((w ·N v) ·N (A ·N 𝐷)) = ((w ·N A) ·N (v ·N 𝐷)))
2721, 26eqeq12d 2051 . . . . 5 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (((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 145 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (((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 139 . . 3 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (([⟨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 6323 . . . . 5 ((x N y N f N) → (x <N y ↔ (f ·N x) <N (f ·N y)))
3130adantl 262 . . . 4 (((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) (x N y N f N)) → (x <N y ↔ (f ·N x) <N (f ·N y)))
3220, 11, 12caovcld 5596 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (z ·N u) N)
3320, 13, 18caovcld 5596 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (B ·N 𝐶) N)
3420, 22, 23caovcld 5596 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (w ·N v) N)
3520, 24, 25caovcld 5596 . . . 4 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (A ·N 𝐷) N)
3631, 32, 33, 34, 15, 35caovord3d 5613 . . 3 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (((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 40 . 2 ((((z N w N) (A N B N)) ((v N u N) (𝐶 N 𝐷 N))) → (([⟨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 6132 1 (((A N B N) (𝐶 N 𝐷 N)) → ([⟨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 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755  (class class class)co 5455  [cec 6040  Ncnpi 6256   ·N cmi 6258
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