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Theorem lt2mulnq 6503
 Description: Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
lt2mulnq (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))

Proof of Theorem lt2mulnq
StepHypRef Expression
1 ltmnqg 6499 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
213expa 1104 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
32adantrr 448 . . . 4 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
4 mulcomnqg 6481 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶))
54ancoms 255 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶))
65ad2ant2r 478 . . . . 5 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → (𝐶 ·Q 𝐴) = (𝐴 ·Q 𝐶))
7 mulcomnqg 6481 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶))
87ancoms 255 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶))
98ad2ant2lr 479 . . . . 5 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → (𝐶 ·Q 𝐵) = (𝐵 ·Q 𝐶))
106, 9breq12d 3777 . . . 4 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶)))
113, 10bitrd 177 . . 3 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → (𝐴 <Q 𝐵 ↔ (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶)))
12 ltmnqg 6499 . . . . . 6 ((𝐶Q𝐷Q𝐵Q) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
13123expa 1104 . . . . 5 (((𝐶Q𝐷Q) ∧ 𝐵Q) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
1413ancoms 255 . . . 4 ((𝐵Q ∧ (𝐶Q𝐷Q)) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
1514adantll 445 . . 3 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → (𝐶 <Q 𝐷 ↔ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
1611, 15anbi12d 442 . 2 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) ↔ ((𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶) ∧ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))))
17 ltsonq 6496 . . 3 <Q Or Q
18 ltrelnq 6463 . . 3 <Q ⊆ (Q × Q)
1917, 18sotri 4720 . 2 (((𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐶) ∧ (𝐵 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷))
2016, 19syl6bi 152 1 (((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393   class class class wbr 3764  (class class class)co 5512  Qcnq 6378   ·Q cmq 6381
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