Theorem mullocpr 7754

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 7658	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmuloc 7749	. . . . . . . 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 2550	. . . . . . 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 1071	. . . . . . . 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 529	. . . . . . . . 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 7559	. . . . . . . . 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 7746	. . . . . . . 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 540	. . . . . . . . 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 7746	. . . . . . . . 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 533	. . . . . . . . . 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 529	. . . . . . . . . 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 539	. . . . . . . . 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 540	. . . . . . . . 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 534	. . . . . . . . . 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 1018	. . . . . . . . . . 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 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)))) ∧ (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 529	. . . . . . . . . 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 7753	. . . . . . . 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 2653	. . . . . . 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 2653	. . . . . 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 1944	. . . 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 2612	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 713   ∧ w3a 1002  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ⟨cop 3669   class class class wbr 4082  ‘cfv 5317  (class class class)co 6000  1^st c1st 6282  2^nd c2nd 6283  Qcnq 7463   ·_Q cmq 7466   <_Q cltq 7468  Pcnp 7474   ·_P cmp 7477

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-imp 7652

This theorem is referenced by:  mulclpr  7755