Theorem cauappcvgprlemopl 7713

Description: Lemma for cauappcvgpr 7729. 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 5929	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2	1	breq1d 4043	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
3	2	rexbidv 2498	. . . . 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 5561	. . . . . 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 7430	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4177	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
8	6	rabex 4177	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
9	7, 8	op1st 6204	. . . . . 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 2217	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
11	3, 10	elrab2 2923	. . . 4 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
12	11	simprbi 275	. . 3 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
13	12	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
14		simprr 531	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
15		ltbtwnnqq 7482	. . . 4 ⊢ ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑡 ∈ Q ((𝑠 +Q 𝑞) <Q 𝑡 ∧ 𝑡 <Q (𝐹‘𝑞)))
16	14, 15	sylib 122	. . 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 535	. . . . . . . 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 274	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
19	18	ad3antlr 493	. . . . . . . 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 7474	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → 𝑞 <Q (𝑞 +Q 𝑠))
21	17, 19, 20	syl2anc 411	. . . . . . 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 7448	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
23	17, 19, 22	syl2anc 411	. . . . . . 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 4059	. . . . . 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 539	. . . . . 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 7465	. . . . . . 7 ⊢ <Q Or Q
27		ltrelnq 7432	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
28	26, 27	sotri 5065	. . . . . 6 ⊢ ((𝑞 <Q (𝑠 +Q 𝑞) ∧ (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
29	24, 25, 28	syl2anc 411	. . . . 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 529	. . . . . 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 7475	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑡 ∈ Q) → (𝑞 <Q 𝑡 ↔ ∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡))
32	17, 30, 31	syl2anc 411	. . . . 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 147	. . . 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 488	. . . . . . . . . 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 488	. . . . . . . . . . . 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 488	. . . . . . . . . . . 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 7448	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
38	35, 36, 37	syl2anc 411	. . . . . . . . . . 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 4043	. . . . . . . . . 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 147	. . . . . . . . 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 110	. . . . . . . . 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 4061	. . . . . . . 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 528	. . . . . . . . 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 7467	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
45	35, 43, 36, 44	syl3anc 1249	. . . . . . . 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 167	. . . . . . 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 540	. . . . . . . . . . 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 488	. . . . . . . . . 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 7448	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
50	36, 43, 49	syl2anc 411	. . . . . . . . . . . 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 2231	. . . . . . . . . . 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 4043	. . . . . . . . . 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 167	. . . . . . . . 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 2546	. . . . . . . . 9 ⊢ ((𝑞 ∈ Q ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
55	36, 53, 54	syl2anc 411	. . . . . . . 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 5929	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
57	56	breq1d 4043	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
58	57	rexbidv 2498	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
59	58, 10	elrab2 2923	. . . . . . . 8 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
60	43, 55, 59	sylanbrc 417	. . . . . . 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 306	. . . . . 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 115	. . . . 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 2599	. . . 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 2619	. 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 2619	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  ⟨cop 3625   class class class wbr 4033  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1^st c1st 6196  Qcnq 7347   +_Q cplq 7349   <_Q cltq 7352

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420

This theorem is referenced by:  cauappcvgprlemrnd  7717