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Theorem mullocpr 7193
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 7097 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmuloc 7188 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
31, 2sylan 278 . . . . . . 7 ((𝐴P𝑞 <Q 𝑟) → ∃𝑑Q𝑢Q (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
4 r2ex 2399 . . . . . . 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 462 . . . . 5 (((𝐴P𝐵P) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
76adantlr 462 . . . 4 ((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
8 simprr3 994 . . . . . . . 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 499 . . . . . . . . 9 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑Q𝑢Q))
10 mulclnq 6998 . . . . . . . . 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 7185 . . . . . . . 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 404 . . . . . . 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 508 . . . . . . . . 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 271 . . . . . . . . 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 7185 . . . . . . . . 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 404 . . . . . . . 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 501 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝐴P𝐵P))
1918ad2antrr 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)))) → (𝐴P𝐵P))
20 simprl 499 . . . . . . . . . 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 474 . . . . . . . . 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 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)))) → (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)))
23 simprrr 508 . . . . . . . . 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 502 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞Q𝑟Q))
2524ad2antrr 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)))) → (𝑞Q𝑟Q))
269ad2antrr 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)))) → (𝑑Q𝑢Q))
27 3simpa 941 . . . . . . . . . . 11 ((𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2827ad2antll 476 . . . . . . . . . 10 (((((𝐴P𝐵P) ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑Q𝑢Q) ∧ (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
2928ad2antrr 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)))) → (𝑑 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴)))
30 simplrl 503 . . . . . . . . . 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 499 . . . . . . . . . 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 301 . . . . . . . . 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 7192 . . . . . . . 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 2496 . . . . . . 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 2496 . . . . . 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 1826 . . . 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 2456 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 665  w3a 925  wex 1427  wcel 1439  wral 2360  wrex 2361  cop 3455   class class class wbr 3853  cfv 5030  (class class class)co 5668  1st c1st 5925  2nd c2nd 5926  Qcnq 6902   ·Q cmq 6905   <Q cltq 6907  Pcnp 6913   ·P cmp 6916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-eprel 4127  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-1o 6197  df-2o 6198  df-oadd 6201  df-omul 6202  df-er 6308  df-ec 6310  df-qs 6314  df-ni 6926  df-pli 6927  df-mi 6928  df-lti 6929  df-plpq 6966  df-mpq 6967  df-enq 6969  df-nqqs 6970  df-plqqs 6971  df-mqqs 6972  df-1nqqs 6973  df-rq 6974  df-ltnqqs 6975  df-enq0 7046  df-nq0 7047  df-0nq0 7048  df-plq0 7049  df-mq0 7050  df-inp 7088  df-imp 7091
This theorem is referenced by:  mulclpr  7194
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