Theorem ltexprlemopu 6793

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 6803. (Contributed by Jim Kingdon, 21-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
ltexprlemopu	⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))

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

Proof of Theorem ltexprlemopu

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . 5 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩
2	1	ltexprlemelu 6789	. . . 4 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
3	2	simprbi 269	. . 3 ⊢ (𝑟 ∈ (2nd ‘𝐶) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
4		19.42v 1827	. . . . . . . 8 ⊢ (∃𝑦(𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ↔ (𝐴<P 𝐵 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))))
5		19.42v 1827	. . . . . . . . 9 ⊢ (∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
6	5	anbi2i 444	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))))
7	4, 6	bitri 182	. . . . . . 7 ⊢ (∃𝑦(𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ↔ (𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))))
8		ltrelpr 6695	. . . . . . . . . . . . . . 15 ⊢ <P ⊆ (P × P)
9	8	brel 4410	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 112	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
11		prop 6665	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prnminu 6679	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
14	12, 13	sylan 277	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
15	14	adantrl 461	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
16	15	adantrl 461	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
17		ltdfpr 6696	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵))))
18	17	biimpd 142	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 → ∃𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵))))
19	9, 18	mpcom 36	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → ∃𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))
20	19	ad2antrr 471	. . . . . . . . . . . 12 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))
21	9	simpld 110	. . . . . . . . . . . . . . . 16 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
22	21	ad2antrr 471	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝐴 ∈ P)
23	22	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝐴 ∈ P)
24		simplrr 502	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
25	24	simpld 110	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 ∈ (1st ‘𝐴))
26	25	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑦 ∈ (1st ‘𝐴))
27		simprrl 505	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑡 ∈ (2nd ‘𝐴))
28		prop 6665	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
29		prltlu 6677	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
30	28, 29	syl3an1 1202	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
31	23, 26, 27, 30	syl3anc 1169	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑦 <Q 𝑡)
32		simplll 499	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝐴<P 𝐵)
33		simprrr 506	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑡 ∈ (1st ‘𝐵))
34		simplrl 501	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (2nd ‘𝐵))
35		prltlu 6677	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
36	12, 35	syl3an1 1202	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
37	32, 33, 34, 36	syl3anc 1169	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑡 <Q 𝑠)
38		ltsonq 6588	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
39		ltrelnq 6555	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
40	38, 39	sotri 4740	. . . . . . . . . . . . 13 ⊢ ((𝑦 <Q 𝑡 ∧ 𝑡 <Q 𝑠) → 𝑦 <Q 𝑠)
41	31, 37, 40	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑡 ∈ Q ∧ (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵)))) → 𝑦 <Q 𝑠)
42	20, 41	rexlimddv 2481	. . . . . . . . . . 11 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 <Q 𝑠)
43		elprnql 6671	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
44	28, 43	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
45	22, 25, 44	syl2anc 403	. . . . . . . . . . . 12 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑦 ∈ Q)
46		elprnqu 6672	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
47	12, 46	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
48	47	ad2ant2r 492	. . . . . . . . . . . 12 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑠 ∈ Q)
49		ltexnqq 6598	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑠 ∈ Q) → (𝑦 <Q 𝑠 ↔ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑠))
50	45, 48, 49	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (𝑦 <Q 𝑠 ↔ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑠))
51	42, 50	mpbid 145	. . . . . . . . . 10 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑠)
52		simprr 498	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) = 𝑠)
53		simplrr 502	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑠 <Q (𝑦 +Q 𝑟))
54	52, 53	eqbrtrd 3805	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
55		simprl 497	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑞 ∈ Q)
56		simplrl 501	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → 𝑟 ∈ Q)
57	56	adantr 270	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑟 ∈ Q)
58	45	adantr 270	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑦 ∈ Q)
59		ltanqg 6590	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
60	55, 57, 58, 59	syl3anc 1169	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
61	54, 60	mpbird 165	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑞 <Q 𝑟)
62	25	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑦 ∈ (1st ‘𝐴))
63		simplrl 501	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → 𝑠 ∈ (2nd ‘𝐵))
64	52, 63	eqeltrd 2155	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))
65	62, 64	jca 300	. . . . . . . . . . . . 13 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))
66	61, 55, 65	jca32 303	. . . . . . . . . . . 12 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ (𝑞 ∈ Q ∧ (𝑦 +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
67	66	expr 367	. . . . . . . . . . 11 ⊢ ((((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) ∧ 𝑞 ∈ Q) → ((𝑦 +Q 𝑞) = 𝑠 → (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))))))
68	67	reximdva 2463	. . . . . . . . . 10 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → (∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑠 → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))))))
69	51, 68	mpd 13	. . . . . . . . 9 ⊢ (((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 <Q (𝑦 +Q 𝑟))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
70	16, 69	rexlimddv 2481	. . . . . . . 8 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
71	70	eximi 1531	. . . . . . 7 ⊢ (∃𝑦(𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑦∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
72	7, 71	sylbir 133	. . . . . 6 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑦∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
73		rexcom4 2622	. . . . . 6 ⊢ (∃𝑞 ∈ Q ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))) ↔ ∃𝑦∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
74	72, 73	sylibr 132	. . . . 5 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑞 ∈ Q ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
75		19.42v 1827	. . . . . . 7 ⊢ (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
76		19.42v 1827	. . . . . . . 8 ⊢ (∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))))
77	76	anbi2i 444	. . . . . . 7 ⊢ ((𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
78	75, 77	bitri 182	. . . . . 6 ⊢ (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
79	78	rexbii 2373	. . . . 5 ⊢ (∃𝑞 ∈ Q ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
80	74, 79	sylib 120	. . . 4 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
81	1	ltexprlemelu 6789	. . . . . 6 ⊢ (𝑞 ∈ (2nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))))
82	81	anbi2i 444	. . . . 5 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
83	82	rexbii 2373	. . . 4 ⊢ (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
84	80, 83	sylibr 132	. . 3 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
85	3, 84	sylanr2 397	. 2 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
86	85	3impb 1134	1 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475  Pcnp 6481  <_P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-ltnqqs 6543  df-inp 6656  df-iltp 6660

This theorem is referenced by:  ltexprlemrnd  6795