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Theorem mullocpr 7600
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 7504 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmuloc 7595 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
31, 2sylan 283 . . . . . . 7 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
4 r2ex 2510 . . . . . . 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 122 . . . . . 6 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
65adantlr 477 . . . . 5 (((𝐴P𝐵P) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
76adantlr 477 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
8 simprr3 1049 . . . . . . . 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 529 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑Q𝑢Q))
10 mulclnq 7405 . . . . . . . . 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 7592 . . . . . . . 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 411 . . . . . . 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 540 . . . . . . . . 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 276 . . . . . . . . 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 7592 . . . . . . . . 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 411 . . . . . . . 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 533 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝐴P𝐵P))
1918ad2antrr 488 . . . . . . . . 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 529 . . . . . . . . . 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 489 . . . . . . . . 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 539 . . . . . . . . 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 540 . . . . . . . . 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 534 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞Q𝑟Q))
2524ad2antrr 488 . . . . . . . . 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 488 . . . . . . . . 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 996 . . . . . . . . . . 11 ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2827ad2antll 491 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2928ad2antrr 488 . . . . . . . . 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 535 . . . . . . . . . 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 529 . . . . . . . . . 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 306 . . . . . . . . 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 7599 . . . . . . . 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 2612 . . . . . . 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 2612 . . . . . 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 115 . . . . 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 1909 . . . 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 115 . 2 (((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
4039ralrimivva 2572 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 104  wo 709  w3a 980  wex 1503  wcel 2160  wral 2468  wrex 2469  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5896  1st c1st 6163  2nd c2nd 6164  Qcnq 7309   ·Q cmq 7312   <Q cltq 7314  Pcnp 7320   ·P cmp 7323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-imp 7498
This theorem is referenced by:  mulclpr  7601
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