Theorem cauappcvgprlemopl 7647

Description: Lemma for cauappcvgpr 7663. 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 5884	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2	1	breq1d 4015	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
3	2	rexbidv 2478	. . . . 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 5520	. . . . . 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 7364	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4149	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
8	6	rabex 4149	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
9	7, 8	op1st 6149	. . . . . 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 2198	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
11	3, 10	elrab2 2898	. . . 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 7416	. . . 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 7408	. . . . . . . 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 7382	. . . . . . . 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 4031	. . . . . 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 7399	. . . . . . 7 ⊢ <Q Or Q
27		ltrelnq 7366	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
28	26, 27	sotri 5026	. . . . . 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 7409	. . . . . 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 7382	. . . . . . . . . . . 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 4015	. . . . . . . . . 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 4033	. . . . . . . 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 7401	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑟 ∈ Q ∧ 𝑞 ∈ Q) → (𝑠 <Q 𝑟 ↔ (𝑞 +Q 𝑠) <Q (𝑞 +Q 𝑟)))
45	35, 43, 36, 44	syl3anc 1238	. . . . . . . 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 7382	. . . . . . . . . . . . 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 2212	. . . . . . . . . . 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 4015	. . . . . . . . . 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 2526	. . . . . . . . 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 5884	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
57	56	breq1d 4015	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
58	57	rexbidv 2478	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
59	58, 10	elrab2 2898	. . . . . . . 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 2579	. . . 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 2599	. 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 2599	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459  ⟨cop 3597   class class class wbr 4005  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  1^st c1st 6141  Qcnq 7281   +_Q cplq 7283   <_Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354

This theorem is referenced by:  cauappcvgprlemrnd  7651