Theorem ltexprlemloc 6705

Description: Our constructed difference is located. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩

Assertion

Ref	Expression
ltexprlemloc	⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))

Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemloc

Dummy variables 𝑧 𝑤 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 6507	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
2	1	adantl 262	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
3		ltrelpr 6603	. . . . . . . . . 10 ⊢ <P ⊆ (P × P)
4	3	brel 4392	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
5	4	simpld 105	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
6		prop 6573	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 6601	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
8	6, 7	sylan 267	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
9	5, 8	sylan 267	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
10	9	ad2ant2r 478	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
11	4	simprd 107	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
12	11	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → 𝐵 ∈ P)
13	12	ad2antrr 457	. . . . . . . . . . . 12 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → 𝐵 ∈ P)
14		ltanqg 6498	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
15	14	adantl 262	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
16		elprnqu 6580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
17	6, 16	sylan 267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
18	5, 17	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
19	18	adantlr 446	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
20	19	ad2ant2rl 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑦 ∈ Q)
21		elprnql 6579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
22	6, 21	sylan 267	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
23	5, 22	sylan 267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
24	23	adantlr 446	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
25	24	ad2ant2r 478	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
26		simplrl 487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
27		addclnq 6473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 +Q 𝑤) ∈ Q)
28	25, 26, 27	syl2anc 391	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑧 +Q 𝑤) ∈ Q)
29		ltrelnq 6463	. . . . . . . . . . . . . . . . . . 19 ⊢ <Q ⊆ (Q × Q)
30	29	brel 4392	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
31	30	simpld 105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 <Q 𝑟 → 𝑞 ∈ Q)
32	31	adantl 262	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ Q)
33	32	ad2antrr 457	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑞 ∈ Q)
34		addcomnqg 6479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
35	34	adantl 262	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
36	15, 20, 28, 33, 35	caovord2d 5670	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) ↔ (𝑦 +Q 𝑞) <Q ((𝑧 +Q 𝑤) +Q 𝑞)))
37		addassnqg 6480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧ 𝑞 ∈ Q) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
38	25, 26, 33, 37	syl3anc 1135	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
39		addcomnqg 6479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
40	26, 33, 39	syl2anc 391	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
41	40	oveq2d 5528	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑧 +Q (𝑤 +Q 𝑞)) = (𝑧 +Q (𝑞 +Q 𝑤)))
42		simplrr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑞 +Q 𝑤) = 𝑟)
43	42	oveq2d 5528	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑧 +Q (𝑞 +Q 𝑤)) = (𝑧 +Q 𝑟))
44	38, 41, 43	3eqtrd 2076	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q 𝑟))
45	44	breq2d 3776	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → ((𝑦 +Q 𝑞) <Q ((𝑧 +Q 𝑤) +Q 𝑞) ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)))
46	36, 45	bitrd 177	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) ↔ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)))
47	46	biimpa 280	. . . . . . . . . . . 12 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟))
48		prop 6573	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
49		prloc 6589	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
50	48, 49	sylan 267	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑞) <Q (𝑧 +Q 𝑟)) → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
51	13, 47, 50	syl2anc 391	. . . . . . . . . . 11 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) ∧ 𝑦 <Q (𝑧 +Q 𝑤)) → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
52	51	ex 108	. . . . . . . . . 10 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → (𝑦 <Q (𝑧 +Q 𝑤) → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
53	52	anassrs 380	. . . . . . . . 9 ⊢ (((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ 𝑧 ∈ (1st ‘𝐴)) ∧ 𝑦 ∈ (2nd ‘𝐴)) → (𝑦 <Q (𝑧 +Q 𝑤) → ((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
54	53	reximdva 2421	. . . . . . . 8 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ 𝑧 ∈ (1st ‘𝐴)) → (∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤) → ∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
55	54	reximdva 2421	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
56		prml 6575	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑧 ∈ Q 𝑧 ∈ (1st ‘𝐴))
57		rexex 2368	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ Q 𝑧 ∈ (1st ‘𝐴) → ∃𝑧 𝑧 ∈ (1st ‘𝐴))
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ∃𝑧 𝑧 ∈ (1st ‘𝐴))
59		r19.45mv 3315	. . . . . . . . . . 11 ⊢ (∃𝑧 𝑧 ∈ (1st ‘𝐴) → (∃𝑧 ∈ (1st ‘𝐴)(∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
60	5, 58, 59	3syl 17	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → (∃𝑧 ∈ (1st ‘𝐴)(∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
61	60	adantr 261	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → (∃𝑧 ∈ (1st ‘𝐴)(∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
62		prmu 6576	. . . . . . . . . . . . 13 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘𝐴))
63		rexex 2368	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘𝐴) → ∃𝑥 𝑥 ∈ (2nd ‘𝐴))
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ∃𝑥 𝑥 ∈ (2nd ‘𝐴))
65		r19.9rmv 3313	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 𝑥 ∈ (2nd ‘𝐴) → ((𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑦 ∈ (2nd ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
66	65	orbi2d 704	. . . . . . . . . . . . 13 ⊢ (∃𝑥 𝑥 ∈ (2nd ‘𝐴) → ((∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑦 ∈ (2nd ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
67		r19.43 2468	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑦 ∈ (2nd ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
68	66, 67	syl6rbbr 188	. . . . . . . . . . . 12 ⊢ (∃𝑥 𝑥 ∈ (2nd ‘𝐴) → (∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 ⊢ (𝐴<P 𝐵 → (∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
70	69	rexbidv 2327	. . . . . . . . . 10 ⊢ (𝐴<P 𝐵 → (∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)(∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
71	70	adantr 261	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → (∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ ∃𝑧 ∈ (1st ‘𝐴)(∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
72		ibar 285	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ Q → (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
73	72	adantr 261	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
74		ibar 285	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ Q → (∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))))
75	74	adantl 262	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))))
76	73, 75	orbi12d 707	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → ((∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))) ↔ ((𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))) ∨ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))))
77	30, 76	syl 14	. . . . . . . . . . . 12 ⊢ (𝑞 <Q 𝑟 → ((∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))) ↔ ((𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))) ∨ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))))
78		ltexprlem.1	. . . . . . . . . . . . . 14 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
79	78	ltexprlemell 6696	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
80	78	ltexprlemelu 6697	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
81		eleq1 2100	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
82		oveq1 5519	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑟) = (𝑧 +Q 𝑟))
83	82	eleq1d 2106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
84	81, 83	anbi12d 442	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
85	84	cbvexv 1795	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
86	85	anbi2i 430	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
87	80, 86	bitri 173	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
88	79, 87	orbi12i 681	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ ((𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))) ∨ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))))
89	77, 88	syl6rbbr 188	. . . . . . . . . . 11 ⊢ (𝑞 <Q 𝑟 → ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))))
90		df-rex 2312	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
91		df-rex 2312	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
92	90, 91	orbi12i 681	. . . . . . . . . . 11 ⊢ ((∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ∨ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
93	89, 92	syl6bbr 187	. . . . . . . . . 10 ⊢ (𝑞 <Q 𝑟 → ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
94	93	adantl 262	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ ∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
95	61, 71, 94	3bitr4rd 210	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
96	95	adantr 261	. . . . . . 7 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → ((𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)) ↔ ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)((𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ∨ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
97	55, 96	sylibrd 158	. . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤) → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
98	10, 97	mpd 13	. . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))
99	2, 98	rexlimddv 2437	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))
100	99	ex 108	. . 3 ⊢ (𝐴<P 𝐵 → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
101	100	ralrimivw 2393	. 2 ⊢ (𝐴<P 𝐵 → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
102	101	ralrimivw 2393	1 ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   +_Q cplq 6380   <_Q cltq 6383  Pcnp 6389  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iltp 6568

This theorem is referenced by:  ltexprlempr  6706