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Theorem ltdcnq 6552
 Description: Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
ltdcnq ((𝐴Q𝐵Q) → DECID 𝐴 <Q 𝐵)

Proof of Theorem ltdcnq
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqpi 6533 . . . 4 (𝐴Q → ∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ))
2 nqpi 6533 . . . 4 (𝐵Q → ∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q ))
31, 2anim12i 325 . . 3 ((𝐴Q𝐵Q) → (∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )))
4 ee4anv 1825 . . 3 (∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) ↔ (∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ∃𝑧𝑤((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )))
53, 4sylibr 141 . 2 ((𝐴Q𝐵Q) → ∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )))
6 mulclpi 6483 . . . . . . . . 9 ((𝑥N𝑤N) → (𝑥 ·N 𝑤) ∈ N)
7 mulclpi 6483 . . . . . . . . 9 ((𝑦N𝑧N) → (𝑦 ·N 𝑧) ∈ N)
8 ltdcpi 6478 . . . . . . . . 9 (((𝑥 ·N 𝑤) ∈ N ∧ (𝑦 ·N 𝑧) ∈ N) → DECID (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧))
96, 7, 8syl2an 277 . . . . . . . 8 (((𝑥N𝑤N) ∧ (𝑦N𝑧N)) → DECID (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧))
109an42s 531 . . . . . . 7 (((𝑥N𝑦N) ∧ (𝑧N𝑤N)) → DECID (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧))
11 ordpipqqs 6529 . . . . . . . 8 (((𝑥N𝑦N) ∧ (𝑧N𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧)))
1211dcbid 759 . . . . . . 7 (((𝑥N𝑦N) ∧ (𝑧N𝑤N)) → (DECID [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~QDECID (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧)))
1310, 12mpbird 160 . . . . . 6 (((𝑥N𝑦N) ∧ (𝑧N𝑤N)) → DECID [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q )
1413ad2ant2r 486 . . . . 5 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → DECID [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q )
15 breq12 3796 . . . . . . 7 ((𝐴 = [⟨𝑥, 𝑦⟩] ~Q𝐵 = [⟨𝑧, 𝑤⟩] ~Q ) → (𝐴 <Q 𝐵 ↔ [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ))
1615ad2ant2l 485 . . . . . 6 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → (𝐴 <Q 𝐵 ↔ [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ))
1716dcbid 759 . . . . 5 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → (DECID 𝐴 <Q 𝐵DECID [⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ))
1814, 17mpbird 160 . . . 4 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → DECID 𝐴 <Q 𝐵)
1918exlimivv 1792 . . 3 (∃𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → DECID 𝐴 <Q 𝐵)
2019exlimivv 1792 . 2 (∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧N𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q )) → DECID 𝐴 <Q 𝐵)
215, 20syl 14 1 ((𝐴Q𝐵Q) → DECID 𝐴 <Q 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  DECID wdc 753   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ⟨cop 3405   class class class wbr 3791  (class class class)co 5539  [cec 6134  Ncnpi 6427   ·N cmi 6429
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