Theorem ltexprlemloc 7922

Description: Our constructed difference is located. Lemma for ltexpri 7928. (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 7724	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
2	1	adantl 277	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
3		ltrelpr 7820	. . . . . . . . . 10 ⊢ <P ⊆ (P × P)
4	3	brel 4802	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
5	4	simpld 112	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
6		prop 7790	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 7818	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
8	6, 7	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
9	5, 8	sylan 283	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
10	9	ad2ant2r 509	. . . . . 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 114	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
12	11	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → 𝐵 ∈ P)
13	12	ad2antrr 488	. . . . . . . . . . . 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 7715	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
15	14	adantl 277	. . . . . . . . . . . . . . 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 7797	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
17	6, 16	sylan 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
18	5, 17	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
19	18	adantlr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
20	19	ad2ant2rl 511	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑦 ∈ Q)
21		elprnql 7796	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
22	6, 21	sylan 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
23	5, 22	sylan 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
24	23	adantlr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
25	24	ad2ant2r 509	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
26		simplrl 537	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
27		addclnq 7690	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 +Q 𝑤) ∈ Q)
28	25, 26, 27	syl2anc 411	. . . . . . . . . . . . . . 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 7680	. . . . . . . . . . . . . . . . . . 19 ⊢ <Q ⊆ (Q × Q)
30	29	brel 4802	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
31	30	simpld 112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 <Q 𝑟 → 𝑞 ∈ Q)
32	31	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ Q)
33	32	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑞 ∈ Q)
34		addcomnqg 7696	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
35	34	adantl 277	. . . . . . . . . . . . . . 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 6224	. . . . . . . . . . . . . 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 7697	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧ 𝑞 ∈ Q) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
38	25, 26, 33, 37	syl3anc 1274	. . . . . . . . . . . . . . . 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 7696	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
40	26, 33, 39	syl2anc 411	. . . . . . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . . . 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 4121	. . . . . . . . . . . . . 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 188	. . . . . . . . . . . . 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 296	. . . . . . . . . . . 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 7790	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
49		prloc 7806	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 115	. . . . . . . . . 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 400	. . . . . . . . 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 2644	. . . . . . . 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 2644	. . . . . . 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 7792	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑧 ∈ Q 𝑧 ∈ (1st ‘𝐴))
57		rexex 2588	. . . . . . . . . . . 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 3603	. . . . . . . . . . 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 276	. . . . . . . . 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 7793	. . . . . . . . . . . . 13 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘𝐴))
63		rexex 2588	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘𝐴) → ∃𝑥 𝑥 ∈ (2nd ‘𝐴))
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ∃𝑥 𝑥 ∈ (2nd ‘𝐴))
65		r19.43 2701	. . . . . . . . . . . . 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)))
66		r19.9rmv 3601	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 𝑥 ∈ (2nd ‘𝐴) → ((𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑦 ∈ (2nd ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
67	66	orbi2d 798	. . . . . . . . . . . . 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))))
68	65, 67	bitr4id 199	. . . . . . . . . . . 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 2543	. . . . . . . . . 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 276	. . . . . . . . 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		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))}⟩
73	72	ltexprlemell 7913	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
74	72	ltexprlemelu 7914	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
75		eleq1 2295	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
76		oveq1 6057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑟) = (𝑧 +Q 𝑟))
77	76	eleq1d 2301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
78	75, 77	anbi12d 473	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
79	78	cbvexv 1968	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
80	79	anbi2i 457	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
81	74, 80	bitri 184	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
82	73, 81	orbi12i 772	. . . . . . . . . . . 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)))))
83		ibar 301	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ Q → (∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))))
84	83	adantr 276	. . . . . . . . . . . . . 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)))))
85		ibar 301	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ Q → (∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))))
86	85	adantl 277	. . . . . . . . . . . . . 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)))))
87	84, 86	orbi12d 801	. . . . . . . . . . . . 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))))))
88	30, 87	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))))))
89	82, 88	bitr4id 199	. . . . . . . . . . 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 2526	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
91		df-rex 2526	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
92	90, 91	orbi12i 772	. . . . . . . . . . 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	bitr4di 198	. . . . . . . . . 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 277	. . . . . . . . 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 221	. . . . . . . 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 276	. . . . . . 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 169	. . . . . 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 2665	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))
100	99	ex 115	. . 3 ⊢ (𝐴<P 𝐵 → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
101	100	ralrimivw 2616	. 2 ⊢ (𝐴<P 𝐵 → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
102	101	ralrimivw 2616	1 ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524  ⟨cop 3692   class class class wbr 4109  ‘cfv 5352  (class class class)co 6050  1^st c1st 6332  2^nd c2nd 6333  Qcnq 7595   +_Q cplq 7597   <_Q cltq 7600  Pcnp 7606  <_P cltp 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  ltexprlempr  7923