Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmuloclemcalc GIF version

Theorem prmuloclemcalc 6663
 Description: Calculations for prmuloc 6664. (Contributed by Jim Kingdon, 9-Dec-2019.)
Hypotheses
Ref Expression
prmuloclemcalc.ru (𝜑𝑅 <Q 𝑈)
prmuloclemcalc.udp (𝜑𝑈 <Q (𝐷 +Q 𝑃))
prmuloclemcalc.axb (𝜑 → (𝐴 +Q 𝑋) = 𝐵)
prmuloclemcalc.pbrx (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))
prmuloclemcalc.a (𝜑𝐴Q)
prmuloclemcalc.b (𝜑𝐵Q)
prmuloclemcalc.d (𝜑𝐷Q)
prmuloclemcalc.p (𝜑𝑃Q)
prmuloclemcalc.x (𝜑𝑋Q)
Assertion
Ref Expression
prmuloclemcalc (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Proof of Theorem prmuloclemcalc
StepHypRef Expression
1 prmuloclemcalc.axb . . . . . . 7 (𝜑 → (𝐴 +Q 𝑋) = 𝐵)
21oveq2d 5528 . . . . . 6 (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = (𝑈 ·Q 𝐵))
3 prmuloclemcalc.ru . . . . . . . . 9 (𝜑𝑅 <Q 𝑈)
4 ltrelnq 6463 . . . . . . . . . 10 <Q ⊆ (Q × Q)
54brel 4392 . . . . . . . . 9 (𝑅 <Q 𝑈 → (𝑅Q𝑈Q))
63, 5syl 14 . . . . . . . 8 (𝜑 → (𝑅Q𝑈Q))
76simprd 107 . . . . . . 7 (𝜑𝑈Q)
8 prmuloclemcalc.a . . . . . . 7 (𝜑𝐴Q)
9 prmuloclemcalc.x . . . . . . 7 (𝜑𝑋Q)
10 distrnqg 6485 . . . . . . 7 ((𝑈Q𝐴Q𝑋Q) → (𝑈 ·Q (𝐴 +Q 𝑋)) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
117, 8, 9, 10syl3anc 1135 . . . . . 6 (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
122, 11eqtr3d 2074 . . . . 5 (𝜑 → (𝑈 ·Q 𝐵) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
13 prmuloclemcalc.b . . . . . . 7 (𝜑𝐵Q)
14 mulcomnqg 6481 . . . . . . 7 ((𝐵Q𝑈Q) → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
1513, 7, 14syl2anc 391 . . . . . 6 (𝜑 → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
16 prmuloclemcalc.udp . . . . . . . . . 10 (𝜑𝑈 <Q (𝐷 +Q 𝑃))
17 ltmnqi 6501 . . . . . . . . . 10 ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝐵Q) → (𝐵 ·Q 𝑈) <Q (𝐵 ·Q (𝐷 +Q 𝑃)))
1816, 13, 17syl2anc 391 . . . . . . . . 9 (𝜑 → (𝐵 ·Q 𝑈) <Q (𝐵 ·Q (𝐷 +Q 𝑃)))
19 prmuloclemcalc.d . . . . . . . . . 10 (𝜑𝐷Q)
20 prmuloclemcalc.p . . . . . . . . . 10 (𝜑𝑃Q)
21 distrnqg 6485 . . . . . . . . . 10 ((𝐵Q𝐷Q𝑃Q) → (𝐵 ·Q (𝐷 +Q 𝑃)) = ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
2213, 19, 20, 21syl3anc 1135 . . . . . . . . 9 (𝜑 → (𝐵 ·Q (𝐷 +Q 𝑃)) = ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
2318, 22breqtrd 3788 . . . . . . . 8 (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
24 mulcomnqg 6481 . . . . . . . . . . 11 ((𝑃Q𝐵Q) → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
2520, 13, 24syl2anc 391 . . . . . . . . . 10 (𝜑 → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
26 prmuloclemcalc.pbrx . . . . . . . . . 10 (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))
2725, 26eqbrtrrd 3786 . . . . . . . . 9 (𝜑 → (𝐵 ·Q 𝑃) <Q (𝑅 ·Q 𝑋))
28 mulclnq 6474 . . . . . . . . . 10 ((𝐵Q𝐷Q) → (𝐵 ·Q 𝐷) ∈ Q)
2913, 19, 28syl2anc 391 . . . . . . . . 9 (𝜑 → (𝐵 ·Q 𝐷) ∈ Q)
30 ltanqi 6500 . . . . . . . . 9 (((𝐵 ·Q 𝑃) <Q (𝑅 ·Q 𝑋) ∧ (𝐵 ·Q 𝐷) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
3127, 29, 30syl2anc 391 . . . . . . . 8 (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
32 ltsonq 6496 . . . . . . . . 9 <Q Or Q
3332, 4sotri 4720 . . . . . . . 8 (((𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) ∧ ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋))) → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
3423, 31, 33syl2anc 391 . . . . . . 7 (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
35 ltmnqi 6501 . . . . . . . . . 10 ((𝑅 <Q 𝑈𝑋Q) → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
363, 9, 35syl2anc 391 . . . . . . . . 9 (𝜑 → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
376simpld 105 . . . . . . . . . 10 (𝜑𝑅Q)
38 mulcomnqg 6481 . . . . . . . . . 10 ((𝑋Q𝑅Q) → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
399, 37, 38syl2anc 391 . . . . . . . . 9 (𝜑 → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
40 mulcomnqg 6481 . . . . . . . . . 10 ((𝑋Q𝑈Q) → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
419, 7, 40syl2anc 391 . . . . . . . . 9 (𝜑 → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
4236, 39, 413brtr3d 3793 . . . . . . . 8 (𝜑 → (𝑅 ·Q 𝑋) <Q (𝑈 ·Q 𝑋))
43 ltanqi 6500 . . . . . . . 8 (((𝑅 ·Q 𝑋) <Q (𝑈 ·Q 𝑋) ∧ (𝐵 ·Q 𝐷) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
4442, 29, 43syl2anc 391 . . . . . . 7 (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
4532, 4sotri 4720 . . . . . . 7 (((𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) ∧ ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋))) → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
4634, 44, 45syl2anc 391 . . . . . 6 (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
4715, 46eqbrtrrd 3786 . . . . 5 (𝜑 → (𝑈 ·Q 𝐵) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
4812, 47eqbrtrrd 3786 . . . 4 (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
49 mulclnq 6474 . . . . . 6 ((𝑈Q𝐴Q) → (𝑈 ·Q 𝐴) ∈ Q)
507, 8, 49syl2anc 391 . . . . 5 (𝜑 → (𝑈 ·Q 𝐴) ∈ Q)
51 mulclnq 6474 . . . . . 6 ((𝑈Q𝑋Q) → (𝑈 ·Q 𝑋) ∈ Q)
527, 9, 51syl2anc 391 . . . . 5 (𝜑 → (𝑈 ·Q 𝑋) ∈ Q)
53 addcomnqg 6479 . . . . 5 (((𝑈 ·Q 𝐴) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)))
5450, 52, 53syl2anc 391 . . . 4 (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)))
55 addcomnqg 6479 . . . . 5 (((𝐵 ·Q 𝐷) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
5629, 52, 55syl2anc 391 . . . 4 (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
5748, 54, 563brtr3d 3793 . . 3 (𝜑 → ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
58 ltanqg 6498 . . . 4 (((𝑈 ·Q 𝐴) ∈ Q ∧ (𝐵 ·Q 𝐷) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷) ↔ ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷))))
5950, 29, 52, 58syl3anc 1135 . . 3 (𝜑 → ((𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷) ↔ ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷))))
6057, 59mpbird 156 . 2 (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷))
61 mulcomnqg 6481 . . 3 ((𝐵Q𝐷Q) → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
6213, 19, 61syl2anc 391 . 2 (𝜑 → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
6360, 62breqtrd 3788 1 (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))
 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 cplq 6380   ·Q cmq 6381
 Copyright terms: Public domain W3C validator