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Theorem mullocpr 6812
Description: Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both 𝐴 and 𝐵 are positive, not just 𝐴). (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
mullocpr ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
Distinct variable groups:   𝐴,𝑞,𝑟   𝐵,𝑞,𝑟

Proof of Theorem mullocpr
Dummy variables 𝑑 𝑒 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6716 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmuloc 6807 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
31, 2sylan 277 . . . . . . 7 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
4 r2ex 2387 . . . . . . 7 (∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) ↔ ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
53, 4sylib 120 . . . . . 6 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
65adantlr 461 . . . . 5 (((𝐴P𝐵P) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
76adantlr 461 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
8 simprr3 989 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))
9 simprl 498 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑Q𝑢Q))
10 mulclnq 6617 . . . . . . . . 9 ((𝑑Q𝑢Q) → (𝑑 ·Q 𝑢) ∈ Q)
119, 10syl 14 . . . . . . . 8 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ·Q 𝑢) ∈ Q)
12 appdivnq 6804 . . . . . . . 8 (((𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟) ∧ (𝑑 ·Q 𝑢) ∈ Q) → ∃𝑒Q ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
138, 11, 12syl2anc 403 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → ∃𝑒Q ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
14 simprrr 507 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟))
1511adantr 270 . . . . . . . . 9 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ·Q 𝑢) ∈ Q)
16 appdivnq 6804 . . . . . . . . 9 (((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟) ∧ (𝑑 ·Q 𝑢) ∈ Q) → ∃𝑡Q ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
1714, 15, 16syl2anc 403 . . . . . . . 8 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → ∃𝑡Q ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
18 simplll 500 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝐴P𝐵P))
1918ad2antrr 472 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝐴P𝐵P))
20 simprl 498 . . . . . . . . . 10 ((𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟))) → (𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)))
2120ad2antlr 473 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)))
22 simprrl 506 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)))
23 simprrr 507 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟))
24 simpllr 501 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞Q𝑟Q))
2524ad2antrr 472 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑞Q𝑟Q))
269ad2antrr 472 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑Q𝑢Q))
27 3simpa 936 . . . . . . . . . . 11 ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2827ad2antll 475 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2928ad2antrr 472 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
30 simplrl 502 . . . . . . . . . 10 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → 𝑒Q)
31 simprl 498 . . . . . . . . . 10 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → 𝑡Q)
3230, 31jca 300 . . . . . . . . 9 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑒Q𝑡Q))
3319, 21, 22, 23, 25, 26, 29, 32mullocprlem 6811 . . . . . . . 8 (((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
3417, 33rexlimddv 2482 . . . . . . 7 ((((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
3513, 34rexlimddv 2482 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
3635ex 113 . . . . 5 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → (((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
3736exlimdvv 1819 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → (∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
387, 37mpd 13 . . 3 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
3938ex 113 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
4039ralrimivva 2444 1 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  w3a 920  wex 1422  wcel 1434  wral 2349  wrex 2350  cop 3403   class class class wbr 3787  cfv 4926  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6521   ·Q cmq 6524   <Q cltq 6526  Pcnp 6532   ·P cmp 6535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-imp 6710
This theorem is referenced by:  mulclpr  6813
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