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Theorem ltexprlemopl 6432

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6444. (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 6429	. . . 4 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑞) ∈ (1^st ‘B))))
3	2	simprbi 260	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐶) → ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑞) ∈ (1^st ‘B)))
4		19.42v 1764	. . . . . . . 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 1764	. . . . . . . . 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 433	. . . . . . . 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 173	. . . . . . 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 6353	. . . . . . . . . . . . . 14 ⊢ <_P ⊆ (P × P)
9	8	brel 4315	. . . . . . . . . . . . 13 ⊢ (A<_P B → (A ∈ P ∧ B ∈ P))
10	9	simprd 107	. . . . . . . . . . . 12 ⊢ (A<_P B → B ∈ P)
11		prop 6323	. . . . . . . . . . . . 13 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
12		prnmaxl 6336	. . . . . . . . . . . . 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 267	. . . . . . . . . . . 12 ⊢ ((B ∈ P ∧ (y +_Q 𝑞) ∈ (1^st ‘B)) → ∃𝑠 ∈ (1^st ‘B)(y +_Q 𝑞) <_Q 𝑠)
14	10, 13	sylan 267	. . . . . . . . . . 11 ⊢ ((A<_P B ∧ (y +_Q 𝑞) ∈ (1^st ‘B)) → ∃𝑠 ∈ (1^st ‘B)(y +_Q 𝑞) <_Q 𝑠)
15	14	adantrl 450	. . . . . . . . . 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 450	. . . . . . . . 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 105	. . . . . . . . . . . . . . 15 ⊢ (A<_P B → A ∈ P)
18	17	ad2antrr 460	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . 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 105	. . . . . . . . . . . . . 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 6323	. . . . . . . . . . . . . . 15 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
22		elprnqu 6330	. . . . . . . . . . . . . . 15 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ y ∈ (2^nd ‘A)) → y ∈ Q)
23	21, 22	sylan 267	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ y ∈ (2^nd ‘A)) → y ∈ Q)
24	18, 20, 23	syl2anc 393	. . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 6259	. . . . . . . . . . . . 13 ⊢ ((y ∈ Q ∧ 𝑞 ∈ Q) → y <_Q (y +_Q 𝑞))
27	24, 25, 26	syl2anc 393	. . . . . . . . . . . 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 472	. . . . . . . . . . . 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 6251	. . . . . . . . . . . . 13 ⊢ <_Q Or Q
30		ltrelnq 6218	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
31	29, 30	sotri 4643	. . . . . . . . . . . 12 ⊢ ((y <_Q (y +_Q 𝑞) ∧ (y +_Q 𝑞) <_Q 𝑠) → y <_Q 𝑠)
32	27, 28, 31	syl2anc 393	. . . . . . . . . . 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 460	. . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 6329	. . . . . . . . . . . . . 14 ⊢ ((⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P ∧ 𝑠 ∈ (1^st ‘B)) → 𝑠 ∈ Q)
36	11, 35	sylan 267	. . . . . . . . . . . . 13 ⊢ ((B ∈ P ∧ 𝑠 ∈ (1^st ‘B)) → 𝑠 ∈ Q)
37	33, 34, 36	syl2anc 393	. . . . . . . . . . . 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 6260	. . . . . . . . . . . 12 ⊢ ((y ∈ Q ∧ 𝑠 ∈ Q) → (y <_Q 𝑠 ↔ ∃𝑟 ∈ Q (y +_Q 𝑟) = 𝑠))
39	24, 37, 38	syl2anc 393	. . . . . . . . . . 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 135	. . . . . . . . . 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 476	. . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . 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 3760	. . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . . 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 6253	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ y ∈ Q) → (𝑞 <_Q 𝑟 ↔ (y +_Q 𝑞) <_Q (y +_Q 𝑟)))
48	44, 45, 46, 47	syl3anc 1119	. . . . . . . . . . . . . 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 156	. . . . . . . . . . . . 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 261	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . 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 2092	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 293	. . . . . . . . . . . 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 357	. . . . . . . . . . 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 2395	. . . . . . . . . 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 2411	. . . . . . . 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 1469	. . . . . . 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 125	. . . . . 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 2550	. . . . . 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 137	. . . . 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 1764	. . . . . . 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 1764	. . . . . . . 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 433	. . . . . . 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 173	. . . . . 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 2305	. . . . 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 127	. . . 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 6429	. . . . . 6 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑟) ∈ (1^st ‘B))))
70	69	anbi2i 433	. . . . 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 2305	. . . 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 137	. . 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 387	. 2 ⊢ ((A<_P B ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
74	73	3impb 1084	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 97 ↔ wb 98 ∧ w3a 871 = wceq 1226 ∃wex 1358 ∈ wcel 1370 ∃wrex 2281 {crab 2284 ⟨cop 3349 class class class wbr 3734 ‘cfv 4825 (class class class)co 5432 1^st c1st 5684 2^nd c2nd 5685 Qcnq 6134 +_Q cplq 6136 <_Q cltq 6139 Pcnp 6145 <_P cltp 6149

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 532 ax-in2 533 ax-io 617 ax-5 1312 ax-7 1313 ax-gen 1314 ax-ie1 1359 ax-ie2 1360 ax-8 1372 ax-10 1373 ax-11 1374 ax-i12 1375 ax-bnd 1376 ax-4 1377 ax-13 1381 ax-14 1382 ax-17 1396 ax-i9 1400 ax-ial 1405 ax-i5r 1406 ax-ext 2000 ax-coll 3842 ax-sep 3845 ax-nul 3853 ax-pow 3897 ax-pr 3914 ax-un 4116 ax-setind 4200 ax-iinf 4234

This theorem depends on definitions: df-bi 110 df-dc 731 df-3or 872 df-3an 873 df-tru 1229 df-fal 1232 df-nf 1326 df-sb 1624 df-eu 1881 df-mo 1882 df-clab 2005 df-cleq 2011 df-clel 2014 df-nfc 2145 df-ne 2184 df-ral 2285 df-rex 2286 df-reu 2287 df-rab 2289 df-v 2533 df-sbc 2738 df-csb 2826 df-dif 2893 df-un 2895 df-in 2897 df-ss 2904 df-nul 3198 df-pw 3332 df-sn 3352 df-pr 3353 df-op 3355 df-uni 3551 df-int 3586 df-iun 3629 df-br 3735 df-opab 3789 df-mpt 3790 df-tr 3825 df-eprel 3996 df-id 4000 df-po 4003 df-iso 4004 df-iord 4048 df-on 4050 df-suc 4053 df-iom 4237 df-xp 4274 df-rel 4275 df-cnv 4276 df-co 4277 df-dm 4278 df-rn 4279 df-res 4280 df-ima 4281 df-iota 4790 df-fun 4827 df-fn 4828 df-f 4829 df-f1 4830 df-fo 4831 df-f1o 4832 df-fv 4833 df-ov 5435 df-oprab 5436 df-mpt2 5437 df-1st 5686 df-2nd 5687 df-recs 5838 df-irdg 5874 df-1o 5912 df-oadd 5916 df-omul 5917 df-er 6013 df-ec 6015 df-qs 6019 df-ni 6158 df-pli 6159 df-mi 6160 df-lti 6161 df-plpq 6197 df-mpq 6198 df-enq 6200 df-nqqs 6201 df-plqqs 6202 df-mqqs 6203 df-1nqqs 6204 df-ltnqqs 6206 df-inp 6314 df-iltp 6318

This theorem is referenced by: ltexprlemrnd 6436

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