Theorem ltexprlemopu 7604

Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7614. (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 7600	. . . 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 1906	. . . . . . . 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 1906	. . . . . . . . 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 7506	. . . . . . . . . . . . . . 15 ⊢ <P ⊆ (P × P)
9	8	brel 4680	. . . . . . . . . . . . . 14 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . . 13 ⊢ (𝐴<P 𝐵 → 𝐵 ∈ P)
11		prop 7476	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<P 𝐵 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
13		prnminu 7490	. . . . . . . . . . . 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 7507	. . . . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
29		prltlu 7488	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
30	28, 29	syl3an1 1271	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1st ‘𝐴) ∧ 𝑡 ∈ (2nd ‘𝐴)) → 𝑦 <Q 𝑡)
31	23, 26, 27, 30	syl3anc 1238	. . . . . . . . . . . . 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 7488	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
36	12, 35	syl3an1 1271	. . . . . . . . . . . . . 14 ⊢ ((𝐴<P 𝐵 ∧ 𝑡 ∈ (1st ‘𝐵) ∧ 𝑠 ∈ (2nd ‘𝐵)) → 𝑡 <Q 𝑠)
37	32, 33, 34, 36	syl3anc 1238	. . . . . . . . . . . . 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 7399	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
39		ltrelnq 7366	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
40	38, 39	sotri 5026	. . . . . . . . . . . . 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 2599	. . . . . . . . . . 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 7482	. . . . . . . . . . . . . 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 7483	. . . . . . . . . . . . . 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 7409	. . . . . . . . . . . 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 4027	. . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <Q 𝑟 ↔ (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟)))
60	55, 57, 58, 59	syl3anc 1238	. . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . 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 2579	. . . . . . . . . 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 2599	. . . . . . . 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 1600	. . . . . . 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 2762	. . . . . 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 1906	. . . . . . 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 1906	. . . . . . . 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 2484	. . . . 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 7600	. . . . . 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 2484	. . . 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 1199	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 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  {crab 2459  ⟨cop 3597   class class class wbr 4005  ‘cfv 5218  (class class class)co 5877  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   +_Q cplq 7283   <_Q cltq 7286  Pcnp 7292  <_P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  ltexprlemrnd  7606