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Theorem nqtri3or 6494
 Description: Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
Assertion
Ref Expression
nqtri3or ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵𝐴 = 𝐵𝐵 <Q 𝐴))

Proof of Theorem nqtri3or
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nqqs 6446 . 2 Q = ((N × N) / ~Q )
2 breq1 3767 . . 3 ([⟨𝑧, 𝑤⟩] ~Q = 𝐴 → ([⟨𝑧, 𝑤⟩] ~Q <Q [⟨𝑢, 𝑣⟩] ~Q𝐴 <Q [⟨𝑢, 𝑣⟩] ~Q ))
3 eqeq1 2046 . . 3 ([⟨𝑧, 𝑤⟩] ~Q = 𝐴 → ([⟨𝑧, 𝑤⟩] ~Q = [⟨𝑢, 𝑣⟩] ~Q𝐴 = [⟨𝑢, 𝑣⟩] ~Q ))
4 breq2 3768 . . 3 ([⟨𝑧, 𝑤⟩] ~Q = 𝐴 → ([⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ [⟨𝑢, 𝑣⟩] ~Q <Q 𝐴))
52, 3, 43orbi123d 1206 . 2 ([⟨𝑧, 𝑤⟩] ~Q = 𝐴 → (([⟨𝑧, 𝑤⟩] ~Q <Q [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑧, 𝑤⟩] ~Q = [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ) ↔ (𝐴 <Q [⟨𝑢, 𝑣⟩] ~Q𝐴 = [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑢, 𝑣⟩] ~Q <Q 𝐴)))
6 breq2 3768 . . 3 ([⟨𝑢, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑢, 𝑣⟩] ~Q𝐴 <Q 𝐵))
7 eqeq2 2049 . . 3 ([⟨𝑢, 𝑣⟩] ~Q = 𝐵 → (𝐴 = [⟨𝑢, 𝑣⟩] ~Q𝐴 = 𝐵))
8 breq1 3767 . . 3 ([⟨𝑢, 𝑣⟩] ~Q = 𝐵 → ([⟨𝑢, 𝑣⟩] ~Q <Q 𝐴𝐵 <Q 𝐴))
96, 7, 83orbi123d 1206 . 2 ([⟨𝑢, 𝑣⟩] ~Q = 𝐵 → ((𝐴 <Q [⟨𝑢, 𝑣⟩] ~Q𝐴 = [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑢, 𝑣⟩] ~Q <Q 𝐴) ↔ (𝐴 <Q 𝐵𝐴 = 𝐵𝐵 <Q 𝐴)))
10 mulclpi 6426 . . . . 5 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) ∈ N)
1110ad2ant2rl 480 . . . 4 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → (𝑧 ·N 𝑣) ∈ N)
12 mulclpi 6426 . . . . 5 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
1312ad2ant2lr 479 . . . 4 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → (𝑤 ·N 𝑢) ∈ N)
14 pitri3or 6420 . . . 4 (((𝑧 ·N 𝑣) ∈ N ∧ (𝑤 ·N 𝑢) ∈ N) → ((𝑧 ·N 𝑣) <N (𝑤 ·N 𝑢) ∨ (𝑧 ·N 𝑣) = (𝑤 ·N 𝑢) ∨ (𝑤 ·N 𝑢) <N (𝑧 ·N 𝑣)))
1511, 13, 14syl2anc 391 . . 3 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ((𝑧 ·N 𝑣) <N (𝑤 ·N 𝑢) ∨ (𝑧 ·N 𝑣) = (𝑤 ·N 𝑢) ∨ (𝑤 ·N 𝑢) <N (𝑧 ·N 𝑣)))
16 ordpipqqs 6472 . . . 4 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ([⟨𝑧, 𝑤⟩] ~Q <Q [⟨𝑢, 𝑣⟩] ~Q ↔ (𝑧 ·N 𝑣) <N (𝑤 ·N 𝑢)))
17 enqeceq 6457 . . . 4 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ([⟨𝑧, 𝑤⟩] ~Q = [⟨𝑢, 𝑣⟩] ~Q ↔ (𝑧 ·N 𝑣) = (𝑤 ·N 𝑢)))
18 ordpipqqs 6472 . . . . . 6 (((𝑢N𝑣N) ∧ (𝑧N𝑤N)) → ([⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑢 ·N 𝑤) <N (𝑣 ·N 𝑧)))
1918ancoms 255 . . . . 5 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ([⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑢 ·N 𝑤) <N (𝑣 ·N 𝑧)))
20 mulcompig 6429 . . . . . . 7 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑢 ·N 𝑤))
2120ad2ant2lr 479 . . . . . 6 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → (𝑤 ·N 𝑢) = (𝑢 ·N 𝑤))
22 mulcompig 6429 . . . . . . 7 ((𝑧N𝑣N) → (𝑧 ·N 𝑣) = (𝑣 ·N 𝑧))
2322ad2ant2rl 480 . . . . . 6 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → (𝑧 ·N 𝑣) = (𝑣 ·N 𝑧))
2421, 23breq12d 3777 . . . . 5 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ((𝑤 ·N 𝑢) <N (𝑧 ·N 𝑣) ↔ (𝑢 ·N 𝑤) <N (𝑣 ·N 𝑧)))
2519, 24bitr4d 180 . . . 4 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ([⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑤 ·N 𝑢) <N (𝑧 ·N 𝑣)))
2616, 17, 253orbi123d 1206 . . 3 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → (([⟨𝑧, 𝑤⟩] ~Q <Q [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑧, 𝑤⟩] ~Q = [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ) ↔ ((𝑧 ·N 𝑣) <N (𝑤 ·N 𝑢) ∨ (𝑧 ·N 𝑣) = (𝑤 ·N 𝑢) ∨ (𝑤 ·N 𝑢) <N (𝑧 ·N 𝑣))))
2715, 26mpbird 156 . 2 (((𝑧N𝑤N) ∧ (𝑢N𝑣N)) → ([⟨𝑧, 𝑤⟩] ~Q <Q [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑧, 𝑤⟩] ~Q = [⟨𝑢, 𝑣⟩] ~Q ∨ [⟨𝑢, 𝑣⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ))
281, 5, 9, 272ecoptocl 6194 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵𝐴 = 𝐵𝐵 <Q 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Ncnpi 6370   ·N cmi 6372
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