Theorem ltexprlemopu 7736

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7746. (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 7732	. . . 4 ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
3	2	simprbi 275	. . 3 ⊢ (𝑟 ∈ (2nd ‘𝐶) → ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))
4		19.42v 1931	. . . . . . . 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 1931	. . . . . . . . 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 457	. . . . . . . 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 184	. . . . . . 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 7638	. . . . . . . . . . . . . . 15 ⊢ <P ⊆ (P × P)
9	8	brel 4735	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
11		prop 7608	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prnminu 7622	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
14	12, 13	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
15	14	adantrl 478	. . . . . . . . . 10 ⊢ ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
16	15	adantrl 478	. . . . . . . . 9 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) → ∃𝑠 ∈ (2nd ‘𝐵)𝑠 <Q (𝑦 +Q 𝑟))
17		ltdfpr 7639	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐴) ∧ 𝑡 ∈ (1st ‘𝐵))))
18	17	biimpd 144	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 112	. . . . . . . . . . . . . . . 16 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P)
22	21	ad2antrr 488	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . . 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 112	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 539	. . . . . . . . . . . . . 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 7608	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
29		prltlu 7620	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
30	28, 29	syl3an1 1283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
31	23, 26, 27, 30	syl3anc 1250	. . . . . . . . . . . . 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 533	. . . . . . . . . . . . . 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 540	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . 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 7620	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
36	12, 35	syl3an1 1283	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
37	32, 33, 34, 36	syl3anc 1250	. . . . . . . . . . . . 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 7531	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
39		ltrelnq 7498	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
40	38, 39	sotri 5087	. . . . . . . . . . . . 13 ⊢ ((𝑦 <Q 𝑡 ∧ 𝑡 <Q 𝑠) → 𝑦 <Q 𝑠)
41	31, 37, 40	syl2anc 411	. . . . . . . . . . . 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 2629	. . . . . . . . . . 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 7614	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
44	28, 43	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴)) → 𝑦 ∈ Q)
45	22, 25, 44	syl2anc 411	. . . . . . . . . . . 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 7615	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
47	12, 46	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴<P 𝐵 ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑠 ∈ Q)
48	47	ad2ant2r 509	. . . . . . . . . . . 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 7541	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑠 ∈ Q) → (𝑦 <Q 𝑠 ↔ ∃𝑞 ∈ Q (𝑦 +Q 𝑞) = 𝑠))
50	45, 48, 49	syl2anc 411	. . . . . . . . . . 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 147	. . . . . . . . . 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 531	. . . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 4073	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 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 7533	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
60	55, 57, 58, 59	syl3anc 1250	. . . . . . . . . . . . . 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 167	. . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . . 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 306	. . . . . . . . . . . . 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 310	. . . . . . . . . . . 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 375	. . . . . . . . . . 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 2609	. . . . . . . . . 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 2629	. . . . . . . 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 1624	. . . . . . 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 135	. . . . . 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 2797	. . . . . 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 134	. . . . 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 1931	. . . . . . 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 1931	. . . . . . . 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 457	. . . . . . 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 184	. . . . . 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 2514	. . . . 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 122	. . . 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 7732	. . . . . 6 ⊢ (𝑞 ∈ (2nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵))))
82	81	anbi2i 457	. . . . 5 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) ↔ (𝑞 <Q 𝑟 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd ‘𝐵)))))
83	82	rexbii 2514	. . . 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 134	. . 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 405	. 2 ⊢ ((𝐴<P 𝐵 ∧ (𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶))) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
86	85	3impb 1202	1 ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  {crab 2489  ⟨cop 3641   class class class wbr 4051  ‘cfv 5280  (class class class)co 5957  1^st c1st 6237  2^nd c2nd 6238  Qcnq 7413   +_Q cplq 7415   <_Q cltq 7418  Pcnp 7424  <_P cltp 7428

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-ltnqqs 7486  df-inp 7599  df-iltp 7603

This theorem is referenced by:  ltexprlemrnd  7738