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Theorem dvcnvrelem1 25905

Description: Lemma for dvcnvre 25907. (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 25786	. . . . . 6 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
3	1, 2	eqsstrrdi 4032	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
4		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	sseldd 3978	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
7	6	rpred 13022	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
8	5, 7	resubcld 11646	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
9	5, 7	readdcld 11247	. . 3 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
10	5, 6	ltsubrpd 13054	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
11	5, 6	ltaddrpd 13055	. . . . 5 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
12	8, 5, 9, 10, 11	lttrd 11379	. . . 4 ⊢ (𝜑 → (𝐶 − 𝑅) < (𝐶 + 𝑅))
13	8, 9, 12	ltled 11366	. . 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 24772	. . . 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 25344	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
19		cncff 24768	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
20	15, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
21	20, 4	ffvelcdmd 7081	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ℝ)
23	8	rexrd 11268	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ^*)
24	9	rexrd 11268	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ^*)
25		lbicc2 13447	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
26	23, 24, 13, 25	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
27	26	adantr 480	. . . . . . . . . 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 11366	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
29	5, 9, 11	ltled 11366	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
30		elicc2 13395	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
32	5, 28, 29, 31	mpbir3and 1339	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
34	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) < 𝐶)
35		isorel 7319	. . . . . . . . . . . . 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 228	. . . . . . . . . . . 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 420	. . . . . . . . . . 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	27	fvresd 6905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) = (𝐹‘(𝐶 − 𝑅)))
41	33	fvresd 6905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) = (𝐹‘𝐶))
42	40, 41	breq12d 5154	. . . . . . . . 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)))
43	39, 42	sylibd 238	. . . . . . . 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)))
44	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐹:𝑋⟶ℝ)
45	44	ffund 6715	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → Fun 𝐹)
46	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
47	44	fdmd 6722	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → dom 𝐹 = 𝑋)
48	46, 47	sseqtrrd 4018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
49		funfvima2 7228	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
50	45, 48, 49	syl2anc 583	. . . . . . . . . . . . 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)))))
51	27, 50	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
52		df-ima 5682	. . . . . . . . . . . . 13 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
53		simprr 770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
54	52, 53	eqtrid 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
55	51, 54	eleqtrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦))
56		elicc2 13395	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)))
57	56	ad2antrl 725	. . . . . . . . . . 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)))
58	55, 57	mpbid 231	. . . . . . . . . 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))
59	58	simp2d 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)))
60		simprll 776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ)
61	14, 26	sseldd 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ 𝑋)
62	20, 61	ffvelcdmd 7081	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
63	62	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
64		lelttr 11308	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
65	60, 63, 22, 64	syl3anc 1368	. . . . . . . . 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)))
66	59, 65	mpand 692	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 − 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
67	43, 66	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)))
68		ubicc2 13448	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
69	23, 24, 13, 68	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
71	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 < (𝐶 + 𝑅))
72		isorel 7319	. . . . . . . . . . . . 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))))
73	72	biimpd 228	. . . . . . . . . . . 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))))
74	73	exp32 420	. . . . . . . . . . 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))))))
75	74	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))))))
76	33, 70, 71, 75	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))))
77		fvex 6898	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ∈ V
78		fvex 6898	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ∈ V
79	77, 78	brcnv 5876	. . . . . . . . . 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))
80	70	fvresd 6905	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) = (𝐹‘(𝐶 + 𝑅)))
81	80, 41	breq12d 5154	. . . . . . . . . 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)))
82	79, 81	bitrid 283	. . . . . . . . 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)))
83	76, 82	sylibd 238	. . . . . . . 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)))
84		funfvima2 7228	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → ((𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
85	45, 48, 84	syl2anc 583	. . . . . . . . . . . . 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)))))
86	70, 85	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
87	86, 54	eleqtrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦))
88		elicc2 13395	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦) ↔ ((𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)))
89	88	ad2antrl 725	. . . . . . . . . . 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)))
90	87, 89	mpbid 231	. . . . . . . . . 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))
91	90	simp2d 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)))
92	14, 69	sseldd 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ 𝑋)
93	20, 92	ffvelcdmd 7081	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
95		lelttr 11308	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℝ) → ((𝑥 ≤ (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶)) → 𝑥 < (𝐹‘𝐶)))
96	60, 94, 22, 95	syl3anc 1368	. . . . . . . . 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)))
97	91, 96	mpand 692	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘(𝐶 + 𝑅)) < (𝐹‘𝐶) → 𝑥 < (𝐹‘𝐶)))
98	83, 97	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)))
99		ax-resscn 11169	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
100	99	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		fss 6728	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
102	20, 99, 101	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
103	14, 3	sstrd 3987	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
104		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
105	104	tgioo2 24674	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106	104, 105	dvres 25795	. . . . . . . . . . . . 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)))))
107	100, 102, 3, 103, 106	syl22anc 836	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
108		iccntr 24692	. . . . . . . . . . . . . 14 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
109	8, 9, 108	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
110	109	reseq2d 5975	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
111	107, 110	eqtrd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
112	111	dmeqd 5899	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
113		dmres 5997	. . . . . . . . . . 11 ⊢ dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹))
114		ioossicc 13416	. . . . . . . . . . . . . 14 ⊢ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))
115	114, 14	sstrid 3988	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ 𝑋)
116	115, 1	sseqtrrd 4018	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹))
117		df-ss 3960	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹) ↔ (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
118	116, 117	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
119	113, 118	eqtrid 2778	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
120	112, 119	eqtrd 2766	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
121		resss 6000	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) ⊆ (ℝ D 𝐹)
122	111, 121	eqsstrdi 4031	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹))
123		rnss 5932	. . . . . . . . . . 11 ⊢ ((ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹) → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
124	122, 123	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ran (ℝ D 𝐹))
125		dvcnvre.z	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
126	124, 125	ssneldd 3980	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
127	8, 9, 17, 120, 126	dvne0 25899	. . . . . . . 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))))))
128	127	adantr 480	. . . . . . 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))))))
129	67, 98, 128	mpjaod 857	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 < (𝐹‘𝐶))
130		isorel 7319	. . . . . . . . . . . . 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))))
131	130	biimpd 228	. . . . . . . . . . . 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))))
132	131	exp32 420	. . . . . . . . . . 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))))))
133	132	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))))))
134	33, 70, 71, 133	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))))
135	41, 80	breq12d 5154	. . . . . . . . 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))))
136	134, 135	sylibd 238	. . . . . . . 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))))
137	90	simp3d 1141	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)
138		simprlr 777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ)
139		ltletr 11310	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 + 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) ∧ (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
140	22, 94, 138, 139	syl3anc 1368	. . . . . . . . 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))
141	137, 140	mpan2d 691	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 + 𝑅)) → (𝐹‘𝐶) < 𝑦))
142	136, 141	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))
143		isorel 7319	. . . . . . . . . . . . 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)))
144	143	biimpd 228	. . . . . . . . . . . 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)))
145	144	exp32 420	. . . . . . . . . . 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)))))
146	145	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)))))
147	27, 33, 34, 146	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)))
148		fvex 6898	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ∈ V
149	148, 77	brcnv 5876	. . . . . . . . . 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)))
150	41, 40	breq12d 5154	. . . . . . . . . 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))))
151	149, 150	bitrid 283	. . . . . . . . 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))))
152	147, 151	sylibd 238	. . . . . . . 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))))
153	58	simp3d 1141	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)
154		ltletr 11310	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ (𝐹‘(𝐶 − 𝑅)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) ∧ (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦) → (𝐹‘𝐶) < 𝑦))
155	22, 63, 138, 154	syl3anc 1368	. . . . . . . . 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))
156	153, 155	mpan2d 691	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐹‘𝐶) < (𝐹‘(𝐶 − 𝑅)) → (𝐹‘𝐶) < 𝑦))
157	152, 156	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))
158	142, 157, 128	mpjaod 857	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) < 𝑦)
159	60	rexrd 11268	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ^*)
160	138	rexrd 11268	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ^*)
161		elioo2 13371	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
162	159, 160, 161	syl2anc 583	. . . . . 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)))
163	22, 129, 158, 162	mpbir3and 1339	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ (𝑥(,)𝑦))
164	54	fveq2d 6889	. . . . . 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)))
165		iccntr 24692	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
166	165	ad2antrl 725	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
167	164, 166	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝑥(,)𝑦))
168	163, 167	eleqtrrd 2830	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
169	168	expr 456	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
170	169	rexlimdvva 3205	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦) → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))))
171	18, 170	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 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ∩ cin 3942 ⊆ wss 3943 class class class wbr 5141 ^◡ccnv 5668 dom cdm 5669 ran crn 5670 ↾ cres 5671 “ cima 5672 Fun wfun 6531 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 Isom wiso 6538 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℝ⁺crp 12980 (,)cioo 13330 [,]cicc 13333 TopOpenctopn 17376 topGenctg 17392 ℂ_fldccnfld 21240 intcnt 22876 –cn→ccncf 24751 D cdv 25747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751

This theorem is referenced by: dvcnvrelem2 25906

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