Theorem mullocpr 7791

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	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (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.

Step	Hyp	Ref	Expression
1		prop 7695	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmuloc 7786	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑞 <Q 𝑟) → ∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
3	1, 2	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → ∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))
4		r2ex 2552	. . . . . . 7 ⊢ (∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) ↔ ∃𝑑∃𝑢((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
5	3, 4	sylib 122	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → ∃𝑑∃𝑢((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
6	5	adantlr 477	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 <Q 𝑟) → ∃𝑑∃𝑢((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
7	6	adantlr 477	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑑∃𝑢((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))))
8		simprr3 1073	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))
9		simprl 531	. . . . . . . . 9 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ Q ∧ 𝑢 ∈ Q))
10		mulclnq 7596	. . . . . . . . 9 ⊢ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) → (𝑑 ·Q 𝑢) ∈ Q)
11	9, 10	syl 14	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ·Q 𝑢) ∈ Q)
12		appdivnq 7783	. . . . . . . 8 ⊢ (((𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟) ∧ (𝑑 ·Q 𝑢) ∈ Q) → ∃𝑒 ∈ Q ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
13	8, 11, 12	syl2anc 411	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → ∃𝑒 ∈ Q ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
14		simprrr 542	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((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))
15	11	adantr 276	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ·Q 𝑢) ∈ Q)
16		appdivnq 7783	. . . . . . . . 9 ⊢ (((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟) ∧ (𝑑 ·Q 𝑢) ∈ Q) → ∃𝑡 ∈ Q ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
17	14, 15, 16	syl2anc 411	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → ∃𝑡 ∈ Q ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))
18		simplll 535	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
19	18	ad2antrr 488	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
20		simprl 531	. . . . . . . . . 10 ⊢ ((𝑒 ∈ Q ∧ ((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)))
21	20	ad2antlr 489	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((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 541	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((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 542	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((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 536	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
25	24	ad2antrr 488	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
26	9	ad2antrr 488	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ Q ∧ 𝑢 ∈ Q))
27		3simpa 1020	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)) → (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴)))
28	27	ad2antll 491	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴)))
29	28	ad2antrr 488	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴)))
30		simplrl 537	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → 𝑒 ∈ Q)
31		simprl 531	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → 𝑡 ∈ Q)
32	30, 31	jca 306	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑡 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑡 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) → (𝑒 ∈ Q ∧ 𝑡 ∈ Q))
33	19, 21, 22, 23, 25, 26, 29, 32	mullocprlem 7790	. . . . . . . 8 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((𝑢 ·Q 𝑞) <Q (𝑒 ·Q (𝑑 ·Q 𝑢)) ∧ (𝑒 ·Q (𝑑 ·Q 𝑢)) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑡 ∈ Q ∧ ((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))))
34	17, 33	rexlimddv 2655	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) ∧ (𝑒 ∈ Q ∧ ((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))))
35	13, 34	rexlimddv 2655	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ ((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟)))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
36	35	ex 115	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → (((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
37	36	exlimdvv 1946	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → (∃𝑑∃𝑢((𝑑 ∈ Q ∧ 𝑢 ∈ Q) ∧ (𝑑 ∈ (1st ‘𝐴) ∧ 𝑢 ∈ (2nd ‘𝐴) ∧ (𝑢 ·Q 𝑞) <Q (𝑑 ·Q 𝑟))) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
38	7, 37	mpd 13	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))
39	38	ex 115	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
40	39	ralrimivva 2614	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 715   ∧ w3a 1004  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  2^nd c2nd 6302  Qcnq 7500   ·_Q cmq 7503   <_Q cltq 7505  Pcnp 7511   ·_P cmp 7514

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-imp 7689

This theorem is referenced by:  mulclpr  7792