Theorem cauappcvgprlem1 7990

Description: Lemma for cauappcvgpr 7993. 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 7741	. . . . 5 ⊢ (𝑅 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
4		simprl 531	. . . . 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 5675	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → (𝐹‘𝑝) = (𝐹‘𝑄))
10		oveq1 6065	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
11	10	oveq2d 6074	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)))
12	9, 11	breq12d 4127	. . . . . . . . . . . . 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 6076	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)))
14	13	breq2d 4126	. . . . . . . . . . . . 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 5675	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝐹‘𝑞) = (𝐹‘𝑥))
17		oveq2 6066	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
18	16, 17	oveq12d 6076	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
19	18	breq2d 4126	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥))))
20	17	oveq2d 6074	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))
21	16, 20	breq12d 4127	. . . . . . . . . . . . 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 2937	. . . . . . . . . . 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 5818	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑥) ∈ Q)
30		addassnqg 7713	. . . . . . . . 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 1274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
32	26, 31	breqtrrd 4142	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) <Q (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥))
33		ltanqg 7731	. . . . . . . . 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 5818	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
36	35	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) ∈ Q)
37		addclnq 7706	. . . . . . . . . 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 7706	. . . . . . . . 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 7712	. . . . . . . . 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 6232	. . . . . . 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 7713	. . . . . . . 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 1274	. . . . . . 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 533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
48	47	oveq2d 6074	. . . . . . 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 7713	. . . . . . . 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 1274	. . . . . . 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 2271	. . . . . 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 4140	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) +Q 𝑥) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
54		oveq2 6066	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑥))
55	16	oveq1d 6073	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
56	54, 55	breq12d 4127	. . . . . 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 2923	. . . . 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 2667	. . 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 7706	. . . . . . . . 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 7989	. . . . . . 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 5679	. . . . . 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 7694	. . . . . . . 8 ⊢ Q ∈ V
67	66	rabex 4261	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
68	66	rabex 4261	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
69	67, 68	op1st 6353	. . . . . 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 2283	. . . . 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 2304	. . . 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 6065	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝑄) → (𝑙 +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑞))
73	72	breq1d 4124	. . . . . . 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 2545	. . . . . 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 2977	. . . . 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 7984	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
80		nqprlu 7878	. . . . 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 7868	. . . 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 7882	. . 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 1005   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  {crab 2526  ⟨cop 3697   class class class wbr 4114  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1^st c1st 6345  Qcnq 7611   +_Q cplq 7613   <_Q cltq 7616  Pcnp 7622   +_P cpp 7624  <_P cltp 7626

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-iltp 7801

This theorem is referenced by:  cauappcvgprlemlim  7992