Theorem cauappcvgprlem1 7657

Description: Lemma for cauappcvgpr 7660. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-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}⟩
cauappcvgprlem.q	⊢ (𝜑 → 𝑄 ∈ Q)
cauappcvgprlem.r	⊢ (𝜑 → 𝑅 ∈ Q)

Assertion

Ref	Expression
cauappcvgprlem1	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,𝑢   𝑄,𝑝,𝑞,𝑙,𝑢   𝑅,𝑝,𝑞,𝑙,𝑢

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

Proof of Theorem cauappcvgprlem1

Dummy variables 𝑓 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgprlem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q)
2		halfnqq 7408	. . . . 5 ⊢ (𝑅 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
4		simprl 529	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑥 ∈ Q)
5		cauappcvgpr.app	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
6	5	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
7		cauappcvgprlem.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ Q)
8	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑄 ∈ Q)
9		fveq2 5515	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → (𝐹‘𝑝) = (𝐹‘𝑄))
10		oveq1 5881	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
11	10	oveq2d 5890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)))
12	9, 11	breq12d 4016	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞))))
13	9, 10	oveq12d 5892	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)))
14	13	breq2d 4015	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹‘𝑞) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞))))
15	12, 14	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑄 → (((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))) ↔ ((𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)))))
16		fveq2 5515	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝐹‘𝑞) = (𝐹‘𝑥))
17		oveq2 5882	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
18	16, 17	oveq12d 5892	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
19	18	breq2d 4015	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥))))
20	17	oveq2d 5890	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))
21	16, 20	breq12d 4016	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹‘𝑥) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥))))
22	19, 21	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑥 → (((𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞))) ↔ ((𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹‘𝑥) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))))
23	15, 22	rspc2v 2854	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ Q ∧ 𝑥 ∈ Q) → (∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹‘𝑥) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))))
24	8, 4, 23	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹‘𝑥) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))))
25	6, 24	mpd 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹‘𝑥) <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥))))
26	25	simpld 112	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
27		cauappcvgpr.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:Q⟶Q)
28	27	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝐹:Q⟶Q)
29	28, 4	ffvelcdmd 5652	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑥) ∈ Q)
30		addassnqg 7380	. . . . . . . . 9 ⊢ (((𝐹‘𝑥) ∈ Q ∧ 𝑄 ∈ Q ∧ 𝑥 ∈ Q) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
31	29, 8, 4, 30	syl3anc 1238	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
32	26, 31	breqtrrd 4031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) <Q (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥))
33		ltanqg 7398	. . . . . . . . 9 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
34	33	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
35	27, 7	ffvelcdmd 5652	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
36	35	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) ∈ Q)
37		addclnq 7373	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) ∈ Q ∧ 𝑄 ∈ Q) → ((𝐹‘𝑥) +Q 𝑄) ∈ Q)
38	29, 8, 37	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑥) +Q 𝑄) ∈ Q)
39		addclnq 7373	. . . . . . . . 9 ⊢ ((((𝐹‘𝑥) +Q 𝑄) ∈ Q ∧ 𝑥 ∈ Q) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
40	38, 4, 39	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
41		addcomnqg 7379	. . . . . . . . 9 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
42	41	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43	34, 36, 40, 4, 42	caovord2d 6043	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) <Q (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) ↔ ((𝐹‘𝑄) +Q 𝑥) <Q ((((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥)))
44	32, 43	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) +Q 𝑥) <Q ((((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥))
45		addassnqg 7380	. . . . . . . 8 ⊢ ((((𝐹‘𝑥) +Q 𝑄) ∈ Q ∧ 𝑥 ∈ Q ∧ 𝑥 ∈ Q) → ((((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹‘𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
46	38, 4, 4, 45	syl3anc 1238	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹‘𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
47		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
48	47	oveq2d 5890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)) = (((𝐹‘𝑥) +Q 𝑄) +Q 𝑅))
49	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑅 ∈ Q)
50		addassnqg 7380	. . . . . . . 8 ⊢ (((𝐹‘𝑥) ∈ Q ∧ 𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
51	29, 8, 49, 50	syl3anc 1238	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
52	46, 48, 51	3eqtrd 2214	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
53	44, 52	breqtrd 4029	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) +Q 𝑥) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
54		oveq2 5882	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑥))
55	16	oveq1d 5889	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
56	54, 55	breq12d 4016	. . . . . 6 ⊢ (𝑞 = 𝑥 → (((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹‘𝑄) +Q 𝑥) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅))))
57	56	rspcev 2841	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ ((𝐹‘𝑄) +Q 𝑥) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅))) → ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)))
58	4, 53, 57	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)))
59	3, 58	rexlimddv 2599	. . 3 ⊢ (𝜑 → ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)))
60		cauappcvgpr.bnd	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
61		cauappcvgpr.lim	. . . . . . . 8 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
62		addclnq 7373	. . . . . . . . 9 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
63	7, 1, 62	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
64	27, 5, 60, 61, 63	cauappcvgprlemladd 7656	. . . . . . 7 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
65	64	fveq2d 5519	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))
66		nqex 7361	. . . . . . . 8 ⊢ Q ∈ V
67	66	rabex 4147	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
68	66	rabex 4147	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
69	67, 68	op1st 6146	. . . . . 6 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))}
70	65, 69	eqtrdi 2226	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))})
71	70	eleq2d 2247	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ (𝐹‘𝑄) ∈ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))}))
72		oveq1 5881	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝑄) → (𝑙 +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑞))
73	72	breq1d 4013	. . . . . . 7 ⊢ (𝑙 = (𝐹‘𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))))
74	73	rexbidv 2478	. . . . . 6 ⊢ (𝑙 = (𝐹‘𝑄) → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))))
75	74	elrab3 2894	. . . . 5 ⊢ ((𝐹‘𝑄) ∈ Q → ((𝐹‘𝑄) ∈ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))} ↔ ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))))
76	35, 75	syl 14	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑄) ∈ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))} ↔ ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))))
77	71, 76	bitrd 188	. . 3 ⊢ (𝜑 → ((𝐹‘𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ∃𝑞 ∈ Q ((𝐹‘𝑄) +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))))
78	59, 77	mpbird 167	. 2 ⊢ (𝜑 → (𝐹‘𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
79	27, 5, 60, 61	cauappcvgprlemcl 7651	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
80		nqprlu 7545	. . . . 5 ⊢ ((𝑄 +Q 𝑅) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P)
81	63, 80	syl 14	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P)
82		addclpr 7535	. . . 4 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
83	79, 81, 82	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
84		nqprl 7549	. . 3 ⊢ (((𝐹‘𝑄) ∈ Q ∧ (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P) → ((𝐹‘𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
85	35, 83, 84	syl2anc 411	. 2 ⊢ (𝜑 → ((𝐹‘𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
86	78, 85	mpbid 147	1 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459  ⟨cop 3595   class class class wbr 4003  ⟶wf 5212  ‘cfv 5216  (class class class)co 5874  1^st c1st 6138  Qcnq 7278   +_Q cplq 7280   <_Q cltq 7283  Pcnp 7289   +_P cpp 7291  <_P cltp 7293

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468

This theorem is referenced by:  cauappcvgprlemlim  7659