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Theorem ltexprlemopl 6565

Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6577. (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 6562	. . . 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 1783	. . . . . . . 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 1783	. . . . . . . . 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 430	. . . . . . . 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 6480	. . . . . . . . . . . . . 14 ⊢ <_P ⊆ (P × P)
9	8	brel 4334	. . . . . . . . . . . . 13 ⊢ (A<_P B → (A ∈ P ∧ B ∈ P))
10	9	simprd 107	. . . . . . . . . . . 12 ⊢ (A<_P B → B ∈ P)
11		prop 6450	. . . . . . . . . . . . 13 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
12		prnmaxl 6463	. . . . . . . . . . . . 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 447	. . . . . . . . . 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 447	. . . . . . . . 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 457	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 6450	. . . . . . . . . . . . . . 15 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
22		elprnqu 6457	. . . . . . . . . . . . . . 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 391	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 6383	. . . . . . . . . . . . 13 ⊢ ((y ∈ Q ∧ 𝑞 ∈ Q) → y <_Q (y +_Q 𝑞))
27	24, 25, 26	syl2anc 391	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6375	. . . . . . . . . . . . 13 ⊢ <_Q Or Q
30		ltrelnq 6342	. . . . . . . . . . . . 13 ⊢ <_Q ⊆ (Q × Q)
31	29, 30	sotri 4662	. . . . . . . . . . . 12 ⊢ ((y <_Q (y +_Q 𝑞) ∧ (y +_Q 𝑞) <_Q 𝑠) → y <_Q 𝑠)
32	27, 28, 31	syl2anc 391	. . . . . . . . . . 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 457	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 6456	. . . . . . . . . . . . . 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 391	. . . . . . . . . . . 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 6384	. . . . . . . . . . . 12 ⊢ ((y ∈ Q ∧ 𝑠 ∈ Q) → (y <_Q 𝑠 ↔ ∃𝑟 ∈ Q (y +_Q 𝑟) = 𝑠))
39	24, 37, 38	syl2anc 391	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 3780	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 6377	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q ∧ y ∈ Q) → (𝑞 <_Q 𝑟 ↔ (y +_Q 𝑞) <_Q (y +_Q 𝑟)))
48	44, 45, 46, 47	syl3anc 1134	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 2111	. . . . . . . . . . . . . 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 2415	. . . . . . . . . 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 2431	. . . . . . . 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 1488	. . . . . . 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 2571	. . . . . 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 1783	. . . . . . 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 1783	. . . . . . . 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 430	. . . . . . 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 2325	. . . . 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 6562	. . . . . 6 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑟) ∈ (1^st ‘B))))
70	69	anbi2i 430	. . . . 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 2325	. . . 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 385	. 2 ⊢ ((A<_P B ∧ (𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶))) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
74	73	3impb 1099	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 884 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 {crab 2304 ⟨cop 3369 class class class wbr 3754 ‘cfv 4844 (class class class)co 5452 1^st c1st 5704 2^nd c2nd 5705 Qcnq 6257 +_Q cplq 6259 <_Q cltq 6262 Pcnp 6268 <_P cltp 6272

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 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3862 ax-sep 3865 ax-nul 3873 ax-pow 3917 ax-pr 3934 ax-un 4135 ax-setind 4219 ax-iinf 4253

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3352 df-sn 3372 df-pr 3373 df-op 3375 df-uni 3571 df-int 3606 df-iun 3649 df-br 3755 df-opab 3809 df-mpt 3810 df-tr 3845 df-eprel 4016 df-id 4020 df-po 4023 df-iso 4024 df-iord 4068 df-on 4070 df-suc 4073 df-iom 4256 df-xp 4293 df-rel 4294 df-cnv 4295 df-co 4296 df-dm 4297 df-rn 4298 df-res 4299 df-ima 4300 df-iota 4809 df-fun 4846 df-fn 4847 df-f 4848 df-f1 4849 df-fo 4850 df-f1o 4851 df-fv 4852 df-ov 5455 df-oprab 5456 df-mpt2 5457 df-1st 5706 df-2nd 5707 df-recs 5858 df-irdg 5894 df-1o 5933 df-oadd 5937 df-omul 5938 df-er 6035 df-ec 6037 df-qs 6041 df-ni 6281 df-pli 6282 df-mi 6283 df-lti 6284 df-plpq 6321 df-mpq 6322 df-enq 6324 df-nqqs 6325 df-plqqs 6326 df-mqqs 6327 df-1nqqs 6328 df-ltnqqs 6330 df-inp 6441 df-iltp 6445

This theorem is referenced by: ltexprlemrnd 6569

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