Theorem cauappcvgprlemopl 8020

Description: Lemma for cauappcvgpr 8036. 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 6095	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2	1	breq1d 4141	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
3	2	rexbidv 2551	. . . . 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 5701	. . . . . 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 7737	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4282	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
8	6	rabex 4282	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
9	7, 8	op1st 6384	. . . . . 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 2259	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
11	3, 10	elrab2 2985	. . . 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 537	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) ∧ (𝑞 ∈ Q ∧ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))) → (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
15		ltbtwnnqq 7789	. . . 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 541	. . . . . . . 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 497	. . . . . . . 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 7781	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → 𝑞 <Q (𝑞 +Q 𝑠))
21	17, 19, 20	syl2anc 415	. . . . . . 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 7755	. . . . . . . 8 ⊢ ((𝑞 ∈ Q ∧ 𝑠 ∈ Q) → (𝑞 +Q 𝑠) = (𝑠 +Q 𝑞))
23	17, 19, 22	syl2anc 415	. . . . . . 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 4157	. . . . . 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 545	. . . . . 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 7772	. . . . . . 7 ⊢ <Q Or Q
27		ltrelnq 7739	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
28	26, 27	sotri 5186	. . . . . 6 ⊢ ((𝑞 <Q (𝑠 +Q 𝑞) ∧ (𝑠 +Q 𝑞) <Q 𝑡) → 𝑞 <Q 𝑡)
29	24, 25, 28	syl2anc 415	. . . . 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 535	. . . . . 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 7782	. . . . . 6 ⊢ ((𝑞 ∈ Q ∧ 𝑡 ∈ Q) → (𝑞 <Q 𝑡 ↔ ∃𝑟 ∈ Q (𝑞 +Q 𝑟) = 𝑡))
32	17, 30, 31	syl2anc 415	. . . . 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 492	. . . . . . . . . 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 492	. . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7755	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 +Q 𝑞) = (𝑞 +Q 𝑠))
38	35, 36, 37	syl2anc 415	. . . . . . . . . . 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 4141	. . . . . . . . . 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 4159	. . . . . . . 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 533	. . . . . . . . 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 7774	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
45	35, 43, 36, 44	syl3anc 1278	. . . . . . . 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 546	. . . . . . . . . . 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 492	. . . . . . . . . 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 7755	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ Q ∧ 𝑟 ∈ Q) → (𝑞 +Q 𝑟) = (𝑟 +Q 𝑞))
50	36, 43, 49	syl2anc 415	. . . . . . . . . . . 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 2273	. . . . . . . . . . 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 4141	. . . . . . . . . 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 2599	. . . . . . . . 9 ⊢ ((𝑞 ∈ Q ∧ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)) → ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞))
55	36, 53, 54	syl2anc 415	. . . . . . . 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 6095	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
57	56	breq1d 4141	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
58	57	rexbidv 2551	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
59	58, 10	elrab2 2985	. . . . . . . 8 ⊢ (𝑟 ∈ (1st ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
60	43, 55, 59	sylanbrc 421	. . . . . . 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 2652	. . . 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 2673	. 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 2673	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4131  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  1^st c1st 6376  Qcnq 7654   +_Q cplq 7656   <_Q cltq 7659

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7678  df-pli 7679  df-mi 7680  df-lti 7681  df-plpq 7718  df-mpq 7719  df-enq 7721  df-nqqs 7722  df-plqqs 7723  df-mqqs 7724  df-1nqqs 7725  df-rq 7726  df-ltnqqs 7727

This theorem is used by:  cauappcvgprlemrnd  8024