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Theorem ltexprlemopl 7969

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7981. (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 7966	. . . 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 1962	. . . . . . . 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 1962	. . . . . . . . 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 461	. . . . . . . 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 7873	. . . . . . . . . . . . . 14 ⊢ <_P ⊆ (P × P)
9	8	brel 4827	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simprd 114	. . . . . . . . . . . 12 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
11		prop 7843	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
12		prnmaxl 7856	. . . . . . . . . . . . 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 482	. . . . . . . . . 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 482	. . . . . . . . 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 492	. . . . . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 7843	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
22		elprnqu 7850	. . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . 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 7775	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑞 ∈ Q) → 𝑦 <_Q (𝑦 +_Q 𝑞))
27	24, 25, 26	syl2anc 415	. . . . . . . . . . . 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 537	. . . . . . . . . . . 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 7766	. . . . . . . . . . . . 13 ⊢ <_Q Or Q
30		ltrelnq 7733	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
31	29, 30	sotri 5183	. . . . . . . . . . . 12 ⊢ ((𝑦 <_Q (𝑦 +_Q 𝑞) ∧ (𝑦 +_Q 𝑞) <_Q 𝑠) → 𝑦 <_Q 𝑠)
32	27, 28, 31	syl2anc 415	. . . . . . . . . . 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 492	. . . . . . . . . . . . 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 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)) → 𝑠 ∈ (1^st ‘𝐵))
35		elprnql 7849	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 7776	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑠 ∈ Q) → (𝑦 <_Q 𝑠 ↔ ∃𝑟 ∈ Q (𝑦 +_Q 𝑟) = 𝑠))
39	24, 37, 38	syl2anc 415	. . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . 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 4158	. . . . . . . . . . . . . 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 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)) → 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 7768	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑦 ∈ Q) → (𝑞 <_Q 𝑟 ↔ (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟)))
48	44, 45, 46, 47	syl3anc 1278	. . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . 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 2315	. . . . . . . . . . . . . 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 2652	. . . . . . . . . 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 2673	. . . . . . . 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 1653	. . . . . . 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 2845	. . . . . 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 1962	. . . . . . 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 1962	. . . . . . . 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 461	. . . . . . 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 2557	. . . . 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 7966	. . . . . 6 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
70	69	anbi2i 461	. . . . 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 2557	. . . 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 409	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
74	73	3impb 1230	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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∃wrex 2529 {crab 2532 ⟨cop 3712 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 1^st c1st 6372 2^nd c2nd 6373 Qcnq 7648 +_Q cplq 7650 <_Q cltq 7653 Pcnp 7659 <_P cltp 7663

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-eprel 4434 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-1o 6687 df-oadd 6691 df-omul 6692 df-er 6807 df-ec 6809 df-qs 6813 df-ni 7672 df-pli 7673 df-mi 7674 df-lti 7675 df-plpq 7712 df-mpq 7713 df-enq 7715 df-nqqs 7716 df-plqqs 7717 df-mqqs 7718 df-1nqqs 7719 df-ltnqqs 7721 df-inp 7834 df-iltp 7838

This theorem is used by: ltexprlemrnd 7973

W3C validator