Theorem cauappcvgprlemopl 6836

Description: Lemma for cauappcvgpr 6852. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩

Assertion

Ref	Expression
cauappcvgprlemopl	⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐿,𝑟,𝑠   𝐴,𝑠,𝑝   𝐹,𝑙,𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑠

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑟,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemopl

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5539	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2	1	breq1d 3795	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
3	2	rexbidv 2369	. . . . 5 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
4		cauappcvgpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
5	4	fveq2i 5201	. . . . . 6 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
6		nqex 6553	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 3922	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
8	6	rabex 3922	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
9	7, 8	op1st 5793	. . . . . 6 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
10	5, 9	eqtri 2101	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
11	3, 10	elrab2 2751	. . . 4 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
12	11	simprbi 269	. . 3 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
13	12	adantl 271	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
14		simprr 498	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
15		ltbtwnnqq 6605	. . . 4 ⊢ ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑡 ∈ Q ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))
16	14, 15	sylib 120	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) → ∃𝑡 ∈ Q ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))
17		simplrl 501	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑞 ∈ Q)
18	11	simplbi 268	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
19	18	ad3antlr 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑠 ∈ Q)
20		ltaddnq 6597	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → 𝑞 <Q (𝑞 +Q 𝑠))
21	17, 19, 20	syl2anc 403	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑞 <Q (𝑞 +Q 𝑠))
22		addcomnqg 6571	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
23	17, 19, 22	syl2anc 403	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
24	21, 23	breqtrd 3809	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑞 <Q (𝑠 +Q 𝑞))
25		simprrl 505	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → (𝑠 +Q 𝑞) <Q 𝑡)
26		ltsonq 6588	. . . . . . 7 ⊢ <Q Or Q
27		ltrelnq 6555	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
28	26, 27	sotri 4740	. . . . . 6 ⊢ ((𝑞 <Q (𝑠 +Q 𝑞) ∧ (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
29	24, 25, 28	syl2anc 403	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑞 <Q 𝑡)
30		simprl 497	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑡 ∈ Q)
31		ltexnqq 6598	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑡 ∈ Q) → (𝑞 <Q 𝑡 ↔ ∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡))
32	17, 30, 31	syl2anc 403	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → (𝑞 <Q 𝑡 ↔ ∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡))
33	29, 32	mpbid 145	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → ∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡)
34	25	ad2antrr 471	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) <Q 𝑡)
35	19	ad2antrr 471	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑠 ∈ Q)
36	17	ad2antrr 471	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑞 ∈ Q)
37		addcomnqg 6571	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
38	35, 36, 37	syl2anc 403	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
39	38	breq1d 3795	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ((𝑠 +Q 𝑞) <Q 𝑡 ↔ (𝑞 +Q 𝑠) <Q 𝑡))
40	34, 39	mpbid 145	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q 𝑡)
41		simpr 108	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = 𝑡)
42	40, 41	breqtrrd 3811	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟))
43		simplr 496	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑟 ∈ Q)
44		ltanqg 6590	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
45	35, 43, 36, 44	syl3anc 1169	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
46	42, 45	mpbird 165	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑠 <Q 𝑟)
47		simprrr 506	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → 𝑡 <Q (𝐹‘𝑞))
48	47	ad2antrr 471	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑡 <Q (𝐹‘𝑞))
49		addcomnqg 6571	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
50	36, 43, 49	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
51	50, 41	eqtr3d 2115	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) = 𝑡)
52	51	breq1d 3795	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ((𝑟 +Q 𝑞) <Q (𝐹‘𝑞) ↔ 𝑡 <Q (𝐹‘𝑞)))
53	48, 52	mpbird 165	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
54		rspe 2412	. . . . . . . . 9 ⊢ ((𝑞 ∈ Q ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
55	36, 53, 54	syl2anc 403	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
56		oveq1 5539	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
57	56	breq1d 3795	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
58	57	rexbidv 2369	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
59	58, 10	elrab2 2751	. . . . . . . 8 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
60	43, 55, 59	sylanbrc 408	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → 𝑟 ∈ (1st ‘𝐿))
61	46, 60	jca 300	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) ∧ (𝑞 +Q 𝑟) = 𝑡) → (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
62	61	ex 113	. . . . 5 ⊢ (((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) ∧ 𝑟 ∈ Q) → ((𝑞 +Q 𝑟) = 𝑡 → (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))))
63	62	reximdva 2463	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → (∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡 → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))))
64	33, 63	mpd 13	. . 3 ⊢ ((((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) ∧ (𝑡 ∈ Q ∧ ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
65	16, 64	rexlimddv 2481	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
66	13, 65	rexlimddv 2481	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543

This theorem is referenced by:  cauappcvgprlemrnd  6840