dvcnvrelem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvcnvrelem1		Structured version Visualization version GIF version

Theorem dvcnvrelem1 23780

Description: Lemma for dvcnvre 23782. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
dvcnvre.f	⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
dvcnvre.d	⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
dvcnvre.z	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
dvcnvre.1	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnvre.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
dvcnvre.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
dvcnvre.s	⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)

**Assertion**
Ref	Expression
dvcnvrelem1	⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))

Proof of Theorem dvcnvrelem1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcnvre.d	. . . . . 6 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
2		dvbsss 23666	. . . . . 6 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
3	1, 2	syl6eqssr 3656	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
4		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	sseldd 3604	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
7	6	rpred 11872	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
8	5, 7	resubcld 10458	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
9	5, 7	readdcld 10069	. . 3 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
10	5, 6	ltsubrpd 11904	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
11	5, 6	ltaddrpd 11905	. . . . 5 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
12	8, 5, 9, 10, 11	lttrd 10198	. . . 4 ⊢ (𝜑 → (𝐶 − 𝑅) < (𝐶 + 𝑅))
13	8, 9, 12	ltled 10185	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ (𝐶 + 𝑅))
14		dvcnvre.s	. . . 4 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
15		dvcnvre.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
16		rescncf 22700	. . . 4 ⊢ (((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 → (𝐹 ∈ (𝑋–cn→ℝ) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)))
17	14, 15, 16	sylc 65	. . 3 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ))
18	8, 9, 13, 17	evthicc2 23229	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
19		cncff 22696	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
20	15, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
21	20, 4	ffvelrnd 6360	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ℝ)
23	8	rexrd 10089	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ^*)
24	9	rexrd 10089	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ^*)
25		lbicc2 12288	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
26	23, 24, 13, 25	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
27	26	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
28	8, 5, 10	ltled 10185	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
29	5, 9, 11	ltled 10185	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
30		elicc2 12238	. . . . . . . . . . . . 13 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
31	8, 9, 30	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
32	5, 28, 29, 31	mpbir3and 1245	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
33	32	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
34	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) < 𝐶)
35		isorel 6576	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
36	35	biimpd 219	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
37	36	exp32 631	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
38	37	com4l 92	. . . . . . . . . 10 ⊢ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
39	27, 33, 34, 38	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
40		fvres 6207	. . . . . . . . . . 11 ⊢ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) = (𝐹‘(𝐶 − 𝑅)))
41	27, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) = (𝐹‘(𝐶 − 𝑅)))
42		fvres 6207	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) = (𝐹‘𝐶))
43	33, 42	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) = (𝐹‘𝐶))
44	41, 43	breq12d 4666	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)))
45	39, 44	sylibd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)))
46	20	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐹:𝑋⟶ℝ)
47		ffun 6048	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
48	46, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → Fun 𝐹)
49	14	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
50		fdm 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
51	46, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → dom 𝐹 = 𝑋)
52	49, 51	sseqtr4d 3642	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
53		funfvima2 6493	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
54	48, 52, 53	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
55	27, 54	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
56		df-ima 5127	. . . . . . . . . . . . 13 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
57		simprr 796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
58	56, 57	syl5eq 2668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
59	55, 58	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦))
60		elicc2 12238	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)))
61	60	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)))
62	59, 61	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦))
63	62	simp2d 1074	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)))
64		simprll 802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ)
65	14, 26	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ 𝑋)
66	20, 65	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
67	66	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
68		lelttr 10128	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
69	64, 67, 22, 68	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
70	63, 69	mpand 711	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
71	45, 70	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → 𝑥 < (𝐹‘𝐶)))
72		ubicc2 12289	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
73	23, 24, 13, 72	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
74	73	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
75	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 < (𝐶 + 𝑅))
76		isorel 6576	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
77	76	biimpd 219	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
78	77	exp32 631	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
79	78	com4l 92	. . . . . . . . . 10 ⊢ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
80	33, 74, 75, 79	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
81		fvex 6201	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ∈ V
82		fvex 6201	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ∈ V
83	81, 82	brcnv 5305	. . . . . . . . . 10 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶))
84		fvres 6207	. . . . . . . . . . . 12 ⊢ ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) = (𝐹‘(𝐶 + 𝑅)))
85	74, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) = (𝐹‘(𝐶 + 𝑅)))
86	85, 43	breq12d 4666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
87	83, 86	syl5bb 272	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
88	80, 87	sylibd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)))
89		funfvima2 6493	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
90	48, 52, 89	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
91	74, 90	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
92	91, 58	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦))
93		elicc2 12238	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)))
94	93	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)))
95	92, 94	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦))
96	95	simp2d 1074	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)))
97	14, 73	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ 𝑋)
98	20, 97	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
99	98	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
100		lelttr 10128	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
101	64, 99, 22, 100	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
102	96, 101	mpand 711	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
103	88, 102	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → 𝑥 < (𝐹‘𝐶)))
104		ax-resscn 9993	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
105	104	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
106		fss 6056	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
107	20, 104, 106	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
108	14, 3	sstrd 3613	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
109		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
110	109	tgioo2 22606	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
111	109, 110	dvres 23675	. . . . . . . . . . . . 13 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ) ∧ (𝑋 ⊆ ℝ ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
112	105, 107, 3, 108, 111	syl22anc 1327	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
113		iccntr 22624	. . . . . . . . . . . . . 14 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
114	8, 9, 113	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
115	114	reseq2d 5396	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
116	112, 115	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
117	116	dmeqd 5326	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
118		dmres 5419	. . . . . . . . . . 11 ⊢ dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹))
119		ioossicc 12259	. . . . . . . . . . . . . 14 ⊢ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))
120	119, 14	syl5ss 3614	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ 𝑋)
121	120, 1	sseqtr4d 3642	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹))
122		df-ss 3588	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹) ↔ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
123	121, 122	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
124	118, 123	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
125	117, 124	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
126		resss 5422	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) ⊆ (ℝ D 𝐹)
127	116, 126	syl6eqss 3655	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹))
128		rnss 5354	. . . . . . . . . . 11 ⊢ ((ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹) → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
129	127, 128	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
130		dvcnvre.z	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
131	129, 130	ssneldd 3606	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
132	8, 9, 17, 125, 131	dvne0 23774	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∨ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
133	132	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∨ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
134	71, 103, 133	mpjaod 396	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 < (𝐹‘𝐶))
135		isorel 6576	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
136	135	biimpd 219	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
137	136	exp32 631	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
138	137	com4l 92	. . . . . . . . . 10 ⊢ (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 < (𝐶 + 𝑅) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))))
139	33, 74, 75, 138	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅))))
140	43, 85	breq12d 4666	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅))))
141	139, 140	sylibd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅))))
142	95	simp3d 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)
143		simprlr 803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ)
144		ltletr 10129	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
145	22, 99, 143, 144	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
146	142, 145	mpan2d 710	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) → (𝐹‘𝐶) < 𝑦))
147	141, 146	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < 𝑦))
148		isorel 6576	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
149	148	biimpd 219	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ∧ 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
150	149	exp32 631	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
151	150	com4l 92	. . . . . . . . . 10 ⊢ ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → ((𝐶 − 𝑅) < 𝐶 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))))
152	27, 33, 34, 151	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶)))
153		fvex 6201	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ∈ V
154	153, 81	brcnv 5305	. . . . . . . . . 10 ⊢ (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)))
155	43, 41	breq12d 4666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
156	154, 155	syl5bb 272	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅))^◡ < ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ↔ (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
157	152, 156	sylibd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅))))
158	62	simp3d 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)
159		ltletr 10129	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
160	22, 67, 143, 159	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
161	158, 160	mpan2d 710	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) → (𝐹‘𝐶) < 𝑦))
162	157, 161	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) Isom < , ^◡ < (((𝐶 − 𝑅)[,](𝐶 + 𝑅)), ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹‘𝐶) < 𝑦))
163	147, 162, 133	mpjaod 396	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) < 𝑦)
164	64	rexrd 10089	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ^*)
165	143	rexrd 10089	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ^*)
166		elioo2 12216	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
167	164, 165, 166	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
168	22, 134, 163, 167	mpbir3and 1245	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ (𝑥(,)𝑦))
169	58	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)))
170		iccntr 22624	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
171	170	ad2antrl 764	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
172	169, 171	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝑥(,)𝑦))
173	168, 172	eleqtrrd 2704	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
174	173	expr 643	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
175	174	rexlimdvva 3038	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
176	18, 175	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℝ⁺crp 11832 (,)cioo 12175 [,]cicc 12178 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 intcnt 20821 –cn→ccncf 22679 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: dvcnvrelem2 23781

W3C validator