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Theorem ltexprlemopl 7784

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7796. (Contributed by Jim Kingdon, 21-Dec-2019.)

**Hypothesis**
Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩

**Assertion**
Ref	Expression
ltexprlemopl	⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))

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

Proof of Theorem ltexprlemopl Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . 5 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
2	1	ltexprlemell 7781	. . . 4 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
3	2	simprbi 275	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐶) → ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))
4		19.42v 1953	. . . . . . . 8 ⊢ (∃𝑦(𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ↔ (𝐴<_P 𝐵 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))))
5		19.42v 1953	. . . . . . . . 9 ⊢ (∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
6	5	anbi2i 457	. . . . . . . 8 ⊢ ((𝐴<_P 𝐵 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ↔ (𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))))
7	4, 6	bitri 184	. . . . . . 7 ⊢ (∃𝑦(𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ↔ (𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))))
8		ltrelpr 7688	. . . . . . . . . . . . . 14 ⊢ <_P ⊆ (P × P)
9	8	brel 4770	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . 12 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
11		prop 7658	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
12		prnmaxl 7671	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) → ∃𝑠 ∈ (1^st ‘𝐵)(𝑦 +_Q 𝑞) <_Q 𝑠)
13	11, 12	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) → ∃𝑠 ∈ (1^st ‘𝐵)(𝑦 +_Q 𝑞) <_Q 𝑠)
14	10, 13	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) → ∃𝑠 ∈ (1^st ‘𝐵)(𝑦 +_Q 𝑞) <_Q 𝑠)
15	14	adantrl 478	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) → ∃𝑠 ∈ (1^st ‘𝐵)(𝑦 +_Q 𝑞) <_Q 𝑠)
16	15	adantrl 478	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑠 ∈ (1^st ‘𝐵)(𝑦 +_Q 𝑞) <_Q 𝑠)
17	9	simpld 112	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝐴 ∈ P)
19		simplrr 536	. . . . . . . . . . . . . . 15 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))
20	19	simpld 112	. . . . . . . . . . . . . 14 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑦 ∈ (2^nd ‘𝐴))
21		prop 7658	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
22		elprnqu 7665	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
23	21, 22	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
24	18, 20, 23	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑦 ∈ Q)
25		simplrl 535	. . . . . . . . . . . . 13 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑞 ∈ Q)
26		ltaddnq 7590	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑞 ∈ Q) → 𝑦 <_Q (𝑦 +_Q 𝑞))
27	24, 25, 26	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑦 <_Q (𝑦 +_Q 𝑞))
28		simprr 531	. . . . . . . . . . . 12 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → (𝑦 +_Q 𝑞) <_Q 𝑠)
29		ltsonq 7581	. . . . . . . . . . . . 13 ⊢ <_Q Or Q
30		ltrelnq 7548	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
31	29, 30	sotri 5123	. . . . . . . . . . . 12 ⊢ ((𝑦 <_Q (𝑦 +_Q 𝑞) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠) → 𝑦 <_Q 𝑠)
32	27, 28, 31	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑦 <_Q 𝑠)
33	10	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝐵 ∈ P)
34		simprl 529	. . . . . . . . . . . . 13 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑠 ∈ (1^st ‘𝐵))
35		elprnql 7664	. . . . . . . . . . . . . 14 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑠 ∈ (1^st ‘𝐵)) → 𝑠 ∈ Q)
36	11, 35	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ 𝑠 ∈ (1^st ‘𝐵)) → 𝑠 ∈ Q)
37	33, 34, 36	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → 𝑠 ∈ Q)
38		ltexnqq 7591	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑠 ∈ Q) → (𝑦 <_Q 𝑠 ↔ ∃𝑟 ∈ Q (𝑦 +_Q 𝑟) = 𝑠))
39	24, 37, 38	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → (𝑦 <_Q 𝑠 ↔ ∃𝑟 ∈ Q (𝑦 +_Q 𝑟) = 𝑠))
40	32, 39	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → ∃𝑟 ∈ Q (𝑦 +_Q 𝑟) = 𝑠)
41		simplrr 536	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑦 +_Q 𝑞) <_Q 𝑠)
42		simprr 531	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑦 +_Q 𝑟) = 𝑠)
43	41, 42	breqtrrd 4110	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
44	25	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑞 ∈ Q)
45		simprl 529	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑟 ∈ Q)
46	24	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑦 ∈ Q)
47		ltanqg 7583	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <_Q 𝑟 ↔ (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟)))
48	44, 45, 46, 47	syl3anc 1271	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑞 <_Q 𝑟 ↔ (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟)))
49	43, 48	mpbird 167	. . . . . . . . . . . . 13 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑞 <_Q 𝑟)
50	20	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑦 ∈ (2^nd ‘𝐴))
51		simplrl 535	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → 𝑠 ∈ (1^st ‘𝐵))
52	42, 51	eqeltrd 2306	. . . . . . . . . . . . . 14 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))
53	50, 52	jca 306	. . . . . . . . . . . . 13 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))
54	49, 45, 53	jca32 310	. . . . . . . . . . . 12 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (𝑦 +_Q 𝑟) = 𝑠)) → (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
55	54	expr 375	. . . . . . . . . . 11 ⊢ ((((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) ∧ 𝑟 ∈ Q) → ((𝑦 +_Q 𝑟) = 𝑠 → (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))))
56	55	reximdva 2632	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → (∃𝑟 ∈ Q (𝑦 +_Q 𝑟) = 𝑠 → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))))
57	40, 56	mpd 13	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) ∧ (𝑠 ∈ (1^st ‘𝐵) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
58	16, 57	rexlimddv 2653	. . . . . . . 8 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
59	58	eximi 1646	. . . . . . 7 ⊢ (∃𝑦(𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑦∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
60	7, 59	sylbir 135	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑦∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
61		rexcom4 2823	. . . . . 6 ⊢ (∃𝑟 ∈ Q ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ ∃𝑦∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
62	60, 61	sylibr 134	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑟 ∈ Q ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
63		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ (𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
64		19.42v 1953	. . . . . . . 8 ⊢ (∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
65	64	anbi2i 457	. . . . . . 7 ⊢ ((𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
66	63, 65	bitri 184	. . . . . 6 ⊢ (∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
67	66	rexbii 2537	. . . . 5 ⊢ (∃𝑟 ∈ Q ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
68	62, 67	sylib 122	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
69	1	ltexprlemell 7781	. . . . . 6 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
70	69	anbi2i 457	. . . . 5 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) ↔ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
71	70	rexbii 2537	. . . 4 ⊢ (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
72	68, 71	sylibr 134	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
73	3, 72	sylanr2 405	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
74	73	3impb 1223	1 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∃wrex 2509 {crab 2512 ⟨cop 3669 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 1^st c1st 6282 2^nd c2nd 6283 Qcnq 7463 +_Q cplq 7465 <_Q cltq 7468 Pcnp 7474 <_P cltp 7478

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-eprel 4379 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-1o 6560 df-oadd 6564 df-omul 6565 df-er 6678 df-ec 6680 df-qs 6684 df-ni 7487 df-pli 7488 df-mi 7489 df-lti 7490 df-plpq 7527 df-mpq 7528 df-enq 7530 df-nqqs 7531 df-plqqs 7532 df-mqqs 7533 df-1nqqs 7534 df-ltnqqs 7536 df-inp 7649 df-iltp 7653

This theorem is referenced by: ltexprlemrnd 7788

W3C validator