Theorem ltexprlemopu 7715

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7725. (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 7711	. . . 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 1929	. . . . . . . 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 1929	. . . . . . . . 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 7617	. . . . . . . . . . . . . . 15 ⊢ <P ⊆ (P × P)
9	8	brel 4726	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
11		prop 7587	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prnminu 7601	. . . . . . . . . . . 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 7618	. . . . . . . . . . . . . . 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 7587	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
29		prltlu 7599	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
30	28, 29	syl3an1 1282	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
31	23, 26, 27, 30	syl3anc 1249	. . . . . . . . . . . . 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 7599	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
36	12, 35	syl3an1 1282	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
37	32, 33, 34, 36	syl3anc 1249	. . . . . . . . . . . . 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 7510	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
39		ltrelnq 7477	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
40	38, 39	sotri 5077	. . . . . . . . . . . . 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 2627	. . . . . . . . . . 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 7593	. . . . . . . . . . . . . 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 7594	. . . . . . . . . . . . . 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 7520	. . . . . . . . . . . 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 4065	. . . . . . . . . . . . . 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 7512	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
60	55, 57, 58, 59	syl3anc 1249	. . . . . . . . . . . . . 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 2281	. . . . . . . . . . . . . 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 2607	. . . . . . . . . 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 2627	. . . . . . . 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 1622	. . . . . . 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 2794	. . . . . 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 1929	. . . . . . 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 1929	. . . . . . . 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 2512	. . . . 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 7711	. . . . . 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 2512	. . . 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 1201	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 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∃wrex 2484  {crab 2487  ⟨cop 3635   class class class wbr 4043  ‘cfv 5270  (class class class)co 5943  1^st c1st 6223  2^nd c2nd 6224  Qcnq 7392   +_Q cplq 7394   <_Q cltq 7397  Pcnp 7403  <_P cltp 7407

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-ltnqqs 7465  df-inp 7578  df-iltp 7582

This theorem is referenced by:  ltexprlemrnd  7717