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Theorem mullocpr 7391
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 7295 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmuloc 7386 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
31, 2sylan 281 . . . . . . 7 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
4 r2ex 2455 . . . . . . 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 121 . . . . . 6 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
65adantlr 468 . . . . 5 (((𝐴P𝐵P) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
76adantlr 468 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
8 simprr3 1031 . . . . . . . 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 520 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑Q𝑢Q))
10 mulclnq 7196 . . . . . . . . 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 7383 . . . . . . . 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 408 . . . . . . 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 529 . . . . . . . . 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 274 . . . . . . . . 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 7383 . . . . . . . . 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 408 . . . . . . . 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 522 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝐴P𝐵P))
1918ad2antrr 479 . . . . . . . . 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 520 . . . . . . . . . 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 480 . . . . . . . . 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 528 . . . . . . . . 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 529 . . . . . . . . 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 523 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞Q𝑟Q))
2524ad2antrr 479 . . . . . . . . 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 479 . . . . . . . . 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 978 . . . . . . . . . . 11 ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2827ad2antll 482 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2928ad2antrr 479 . . . . . . . . 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 524 . . . . . . . . . 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 520 . . . . . . . . . 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 304 . . . . . . . . 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 7390 . . . . . . . 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 2554 . . . . . . 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 2554 . . . . . 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 114 . . . . 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 1869 . . . 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 114 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
4039ralrimivva 2514 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 103  wo 697  w3a 962  wex 1468  wcel 1480  wral 2416  wrex 2417  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   ·Q cmq 7103   <Q cltq 7105  Pcnp 7111   ·P cmp 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-imp 7289
This theorem is referenced by:  mulclpr  7392
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