Theorem cauappcvgprlem1 8020

Description: Lemma for cauappcvgpr 8023. 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 7771	. . . . 5 ⊢ (𝑅 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑅)
4		simprl 535	. . . . 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 5693	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → (𝐹‘𝑝) = (𝐹‘𝑄))
10		oveq1 6086	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
11	10	oveq2d 6095	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)))
12	9, 11	breq12d 4141	. . . . . . . . . . . . 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 6097	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑄 → ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)))
14	13	breq2d 4140	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 5693	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝐹‘𝑞) = (𝐹‘𝑥))
17		oveq2 6087	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
18	16, 17	oveq12d 6097	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
19	18	breq2d 4140	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹‘𝑄) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥))))
20	17	oveq2d 6095	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹‘𝑄) +Q (𝑄 +Q 𝑥)))
21	16, 20	breq12d 4141	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 2943	. . . . . . . . . . 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 415	. . . . . . . . . 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 5838	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑥) ∈ Q)
30		addassnqg 7743	. . . . . . . . 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 1278	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑥)))
32	26, 31	breqtrrd 4156	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) <Q (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥))
33		ltanqg 7761	. . . . . . . . 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 5838	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑄) ∈ Q)
36	35	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹‘𝑄) ∈ Q)
37		addclnq 7736	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) ∈ Q ∧ 𝑄 ∈ Q) → ((𝐹‘𝑥) +Q 𝑄) ∈ Q)
38	29, 8, 37	syl2anc 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑥) +Q 𝑄) ∈ Q)
39		addclnq 7736	. . . . . . . . 9 ⊢ ((((𝐹‘𝑥) +Q 𝑄) ∈ Q ∧ 𝑥 ∈ Q) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
40	38, 4, 39	syl2anc 415	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹‘𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
41		addcomnqg 7742	. . . . . . . . 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 6253	. . . . . . 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 7743	. . . . . . . 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 1278	. . . . . . 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 537	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
48	47	oveq2d 6095	. . . . . . 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 7743	. . . . . . . 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 1278	. . . . . . 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 2275	. . . . . 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 4154	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹‘𝑄) +Q 𝑥) <Q ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
54		oveq2 6087	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑄) +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑥))
55	16	oveq1d 6094	. . . . . . 7 ⊢ (𝑞 = 𝑥 → ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹‘𝑥) +Q (𝑄 +Q 𝑅)))
56	54, 55	breq12d 4141	. . . . . 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 2929	. . . . 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 415	. . . 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 2673	. . 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 7736	. . . . . . . . 9 ⊢ ((𝑄 ∈ Q ∧ 𝑅 ∈ Q) → (𝑄 +Q 𝑅) ∈ Q)
63	7, 1, 62	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
64	27, 5, 60, 61, 63	cauappcvgprlemladd 8019	. . . . . . 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 5697	. . . . . 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 7724	. . . . . . . 8 ⊢ Q ∈ V
67	66	rabex 4278	. . . . . . 7 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
68	66	rabex 4278	. . . . . . 7 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
69	67, 68	op1st 6374	. . . . . 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 2287	. . . . 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 2308	. . . 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 6086	. . . . . . . 8 ⊢ (𝑙 = (𝐹‘𝑄) → (𝑙 +Q 𝑞) = ((𝐹‘𝑄) +Q 𝑞))
73	72	breq1d 4138	. . . . . . 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 2551	. . . . . 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 2983	. . . . 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 8014	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
80		nqprlu 7908	. . . . 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 7898	. . . 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 415	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
84		nqprl 7912	. . 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 415	. 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 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3711   class class class wbr 4128  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  Qcnq 7641   +_Q cplq 7643   <_Q cltq 7646  Pcnp 7652   +_P cpp 7654  <_P cltp 7656

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-iplp 7829  df-iltp 7831

This theorem is referenced by:  cauappcvgprlemlim  8022