Theorem ltexprlemopu 7918

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7928. (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 7914	. . . 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 1956	. . . . . . . 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 1956	. . . . . . . . 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 7820	. . . . . . . . . . . . . . 15 ⊢ <P ⊆ (P × P)
9	8	brel 4802	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
11		prop 7790	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prnminu 7804	. . . . . . . . . . . 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 7821	. . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . 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 7790	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
29		prltlu 7802	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
30	28, 29	syl3an1 1307	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
31	23, 26, 27, 30	syl3anc 1274	. . . . . . . . . . . . 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 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)))) → 𝐴<P 𝐵)
33		simprrr 542	. . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . 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 7802	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
36	12, 35	syl3an1 1307	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
37	32, 33, 34, 36	syl3anc 1274	. . . . . . . . . . . . 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 7713	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
39		ltrelnq 7680	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
40	38, 39	sotri 5158	. . . . . . . . . . . . 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 2665	. . . . . . . . . . 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 7796	. . . . . . . . . . . . . 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 7797	. . . . . . . . . . . . . 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 7723	. . . . . . . . . . . 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 533	. . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . 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 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)) → 𝑞 ∈ Q)
56		simplrl 537	. . . . . . . . . . . . . . . 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 7715	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
60	55, 57, 58, 59	syl3anc 1274	. . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . 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 2309	. . . . . . . . . . . . . 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 2644	. . . . . . . . . 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 2665	. . . . . . . 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 1649	. . . . . . 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 2837	. . . . . 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 1956	. . . . . . 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 1956	. . . . . . . 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 2549	. . . . 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 7914	. . . . . 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 2549	. . . 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 1226	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 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃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-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-ltnqqs 7668  df-inp 7781  df-iltp 7785

This theorem is referenced by:  ltexprlemrnd  7920