Theorem ltexprlemloc 7548

Description: Our constructed difference is located. Lemma for ltexpri 7554. (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 7350	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
2	1	adantl 275	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → ∃𝑤 ∈ Q (𝑞 +Q 𝑤) = 𝑟)
3		ltrelpr 7446	. . . . . . . . . 10 ⊢ <P ⊆ (P × P)
4	3	brel 4656	. . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
5	4	simpld 111	. . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
6		prop 7416	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		prarloc 7444	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
8	6, 7	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
9	5, 8	sylan 281	. . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝑤 ∈ Q) → ∃𝑧 ∈ (1st ‘𝐴)∃𝑦 ∈ (2nd ‘𝐴)𝑦 <Q (𝑧 +Q 𝑤))
10	9	ad2ant2r 501	. . . . . 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 113	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
12	11	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) → 𝐵 ∈ P)
13	12	ad2antrr 480	. . . . . . . . . . . 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 7341	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
15	14	adantl 275	. . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
17	6, 16	sylan 281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
18	5, 17	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴<P 𝐵 ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
19	18	adantlr 469	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑦 ∈ (2nd ‘𝐴)) → 𝑦 ∈ Q)
20	19	ad2ant2rl 503	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑦 ∈ Q)
21		elprnql 7422	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
22	6, 21	sylan 281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
23	5, 22	sylan 281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴<P 𝐵 ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
24	23	adantlr 469	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ 𝑧 ∈ (1st ‘𝐴)) → 𝑧 ∈ Q)
25	24	ad2ant2r 501	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
26		simplrl 525	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
27		addclnq 7316	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 +Q 𝑤) ∈ Q)
28	25, 26, 27	syl2anc 409	. . . . . . . . . . . . . . 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 7306	. . . . . . . . . . . . . . . . . . 19 ⊢ <Q ⊆ (Q × Q)
30	29	brel 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
31	30	simpld 111	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 <Q 𝑟 → 𝑞 ∈ Q)
32	31	adantl 275	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → 𝑞 ∈ Q)
33	32	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) ∧ (𝑤 ∈ Q ∧ (𝑞 +Q 𝑤) = 𝑟)) ∧ (𝑧 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐴))) → 𝑞 ∈ Q)
34		addcomnqg 7322	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
35	34	adantl 275	. . . . . . . . . . . . . . 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 6011	. . . . . . . . . . . . . 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 7323	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧ 𝑞 ∈ Q) → ((𝑧 +Q 𝑤) +Q 𝑞) = (𝑧 +Q (𝑤 +Q 𝑞)))
38	25, 26, 33, 37	syl3anc 1228	. . . . . . . . . . . . . . . 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 7322	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q) → (𝑤 +Q 𝑞) = (𝑞 +Q 𝑤))
40	26, 33, 39	syl2anc 409	. . . . . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . . . 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 2202	. . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . . 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 187	. . . . . . . . . . . . 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 294	. . . . . . . . . . . 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 7416	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
49		prloc 7432	. . . . . . . . . . . . 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 281	. . . . . . . . . . . 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 409	. . . . . . . . . . 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 114	. . . . . . . . . 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 398	. . . . . . . . 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 2568	. . . . . . . 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 2568	. . . . . . 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 7418	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑧 ∈ Q 𝑧 ∈ (1st ‘𝐴))
57		rexex 2512	. . . . . . . . . . . 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 3502	. . . . . . . . . . 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 274	. . . . . . . . 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 7419	. . . . . . . . . . . . 13 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘𝐴))
63		rexex 2512	. . . . . . . . . . . . 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 2624	. . . . . . . . . . . . 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 3500	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 𝑥 ∈ (2nd ‘𝐴) → ((𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑦 ∈ (2nd ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
67	66	orbi2d 780	. . . . . . . . . . . . 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 198	. . . . . . . . . . . 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 2467	. . . . . . . . . 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 274	. . . . . . . . 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 7539	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
74	72	ltexprlemelu 7540	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
75		eleq1 2229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1st ‘𝐴) ↔ 𝑧 ∈ (1st ‘𝐴)))
76		oveq1 5849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑟) = (𝑧 +Q 𝑟))
77	76	eleq1d 2235	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
78	75, 77	anbi12d 465	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ (𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
79	78	cbvexv 1906	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
80	79	anbi2i 453	. . . . . . . . . . . . . 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 183	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵))))
82	73, 81	orbi12i 754	. . . . . . . . . . . 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 299	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 299	. . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 198	. . . . . . . . . . 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 2450	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ (2nd ‘𝐴)(𝑦 +Q 𝑞) ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))
91		df-rex 2450	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (1st ‘𝐴)(𝑧 +Q 𝑟) ∈ (2nd ‘𝐵) ↔ ∃𝑧(𝑧 ∈ (1st ‘𝐴) ∧ (𝑧 +Q 𝑟) ∈ (2nd ‘𝐵)))
92	90, 91	orbi12i 754	. . . . . . . . . . 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 197	. . . . . . . . . 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 275	. . . . . . . . 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 220	. . . . . . . 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 274	. . . . . . 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 168	. . . . . 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 2588	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))
100	99	ex 114	. . 3 ⊢ (𝐴<P 𝐵 → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
101	100	ralrimivw 2540	. 2 ⊢ (𝐴<P 𝐵 → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
102	101	ralrimivw 2540	1 ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448  ⟨cop 3579   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221   +_Q cplq 7223   <_Q cltq 7226  Pcnp 7232  <_P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iltp 7411

This theorem is referenced by:  ltexprlempr  7549