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Theorem mullocprlem 7484
 Description: Calculations for mullocpr 7485. (Contributed by Jim Kingdon, 10-Dec-2019.)
Hypotheses
Ref Expression
mullocprlem.ab (𝜑 → (𝐴P𝐵P))
mullocprlem.uqedu (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.edutdu (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.tdudr (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
mullocprlem.qr (𝜑 → (𝑄Q𝑅Q))
mullocprlem.duq (𝜑 → (𝐷Q𝑈Q))
mullocprlem.du (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
mullocprlem.et (𝜑 → (𝐸Q𝑇Q))
Assertion
Ref Expression
mullocprlem (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mullocprlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mullocprlem.uqedu . . . . . . 7 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
2 mullocprlem.et . . . . . . . . 9 (𝜑 → (𝐸Q𝑇Q))
32simpld 111 . . . . . . . 8 (𝜑𝐸Q)
4 mullocprlem.duq . . . . . . . . 9 (𝜑 → (𝐷Q𝑈Q))
54simpld 111 . . . . . . . 8 (𝜑𝐷Q)
64simprd 113 . . . . . . . 8 (𝜑𝑈Q)
7 mulcomnqg 7297 . . . . . . . . 9 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
87adantl 275 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulassnqg 7298 . . . . . . . . 9 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
109adantl 275 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q𝑧Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
113, 5, 6, 8, 10caov13d 6001 . . . . . . 7 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
121, 11breqtrd 3990 . . . . . 6 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13 mullocprlem.qr . . . . . . . 8 (𝜑 → (𝑄Q𝑅Q))
1413simpld 111 . . . . . . 7 (𝜑𝑄Q)
15 mulclnq 7290 . . . . . . . 8 ((𝐷Q𝐸Q) → (𝐷 ·Q 𝐸) ∈ Q)
165, 3, 15syl2anc 409 . . . . . . 7 (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17 ltmnqg 7315 . . . . . . 7 ((𝑄Q ∧ (𝐷 ·Q 𝐸) ∈ Q𝑈Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1814, 16, 6, 17syl3anc 1220 . . . . . 6 (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1912, 18mpbird 166 . . . . 5 (𝜑𝑄 <Q (𝐷 ·Q 𝐸))
2019adantr 274 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21 mullocprlem.ab . . . . . . . 8 (𝜑 → (𝐴P𝐵P))
2221simpld 111 . . . . . . 7 (𝜑𝐴P)
23 mullocprlem.du . . . . . . . 8 (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
2423simpld 111 . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
2522, 24jca 304 . . . . . 6 (𝜑 → (𝐴P𝐷 ∈ (1st𝐴)))
2625adantr 274 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐴P𝐷 ∈ (1st𝐴)))
2721simprd 113 . . . . . 6 (𝜑𝐵P)
2827anim1i 338 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐵P𝐸 ∈ (1st𝐵)))
2914adantr 274 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄Q)
30 mulnqprl 7482 . . . . 5 ((((𝐴P𝐷 ∈ (1st𝐴)) ∧ (𝐵P𝐸 ∈ (1st𝐵))) ∧ 𝑄Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3126, 28, 29, 30syl21anc 1219 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3220, 31mpd 13 . . 3 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)))
3332orcd 723 . 2 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
342simprd 113 . . . . . . 7 (𝜑𝑇Q)
35 mulcomnqg 7297 . . . . . . 7 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
3634, 6, 35syl2anc 409 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37 mullocprlem.tdudr . . . . . . 7 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38 mulclnq 7290 . . . . . . . . . 10 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) ∈ Q)
3934, 6, 38syl2anc 409 . . . . . . . . 9 (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
4013simprd 113 . . . . . . . . 9 (𝜑𝑅Q)
41 ltmnqg 7315 . . . . . . . . 9 (((𝑇 ·Q 𝑈) ∈ Q𝑅Q𝐷Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4239, 40, 5, 41syl3anc 1220 . . . . . . . 8 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4334, 5, 6, 8, 10caov12d 5999 . . . . . . . . 9 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
4443breq1d 3975 . . . . . . . 8 (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4542, 44bitr4d 190 . . . . . . 7 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4637, 45mpbird 166 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
4736, 46eqbrtrrd 3988 . . . . 5 (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
4847adantr 274 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
4923simprd 113 . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
5022, 49jca 304 . . . . . 6 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
5150adantr 274 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐴P𝑈 ∈ (2nd𝐴)))
5227anim1i 338 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐵P𝑇 ∈ (2nd𝐵)))
5340adantr 274 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅Q)
54 mulnqpru 7483 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5551, 52, 53, 54syl21anc 1219 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5648, 55mpd 13 . . 3 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
5756olcd 724 . 2 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
58 mullocprlem.edutdu . . . 4 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
59 mulclnq 7290 . . . . . . 7 ((𝐷Q𝑈Q) → (𝐷 ·Q 𝑈) ∈ Q)
604, 59syl 14 . . . . . 6 (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61 ltmnqg 7315 . . . . . 6 ((𝐸Q𝑇Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
623, 34, 60, 61syl3anc 1220 . . . . 5 (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63 mulcomnqg 7297 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝐸Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
6460, 3, 63syl2anc 409 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65 mulcomnqg 7297 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝑇Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6660, 34, 65syl2anc 409 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6764, 66breq12d 3978 . . . . 5 (𝜑 → (((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇) ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6862, 67bitrd 187 . . . 4 (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6958, 68mpbird 166 . . 3 (𝜑𝐸 <Q 𝑇)
70 prop 7389 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
71 prloc 7405 . . . 4 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7270, 71sylan 281 . . 3 ((𝐵P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7327, 69, 72syl2anc 409 . 2 (𝜑 → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7433, 57, 73mpjaodan 788 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  ‘cfv 5169  (class class class)co 5821  1st c1st 6083  2nd c2nd 6084  Qcnq 7194   ·Q cmq 7197
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