Theorem cauappcvgprlemopl 7620

Description: Lemma for cauappcvgpr 7636. 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 5872	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
2	1	breq1d 4008	. . . . . 6 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
3	2	rexbidv 2476	. . . . 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 5510	. . . . . 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 7337	. . . . . . . 8 ⊢ Q ∈ V
7	6	rabex 4142	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
8	6	rabex 4142	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
9	7, 8	op1st 6137	. . . . . 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 2196	. . . . 5 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
11	3, 10	elrab2 2894	. . . 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 7389	. . . 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 7381	. . . . . . . 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 7355	. . . . . . . 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 4024	. . . . . 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 7372	. . . . . . 7 ⊢ <Q Or Q
27		ltrelnq 7339	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
28	26, 27	sotri 5016	. . . . . 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 7382	. . . . . 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 7355	. . . . . . . . . . . 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 4008	. . . . . . . . . 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 4026	. . . . . . . 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 7374	. . . . . . . . 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 7355	. . . . . . . . . . . . 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 2210	. . . . . . . . . . 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 4008	. . . . . . . . . 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 2524	. . . . . . . . 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 5872	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑟 → (𝑙 +Q 𝑞) = (𝑟 +Q 𝑞))
57	56	breq1d 4008	. . . . . . . . . 10 ⊢ (𝑙 = 𝑟 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
58	57	rexbidv 2476	. . . . . . . . 9 ⊢ (𝑙 = 𝑟 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑟 +Q 𝑞) <Q (𝐹‘𝑞)))
59	58, 10	elrab2 2894	. . . . . . . 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 2577	. . . 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 2597	. 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 2597	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  {crab 2457  ⟨cop 3592   class class class wbr 3998  ⟶wf 5204  ‘cfv 5208  (class class class)co 5865  1^st c1st 6129  Qcnq 7254   +_Q cplq 7256   <_Q cltq 7259

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327

This theorem is referenced by:  cauappcvgprlemrnd  7624