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Theorem dvcnvrelem1 25955

Description: Lemma for dvcnvre 25957. (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 25836	. . . . . 6 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
3	1, 2	eqsstrrdi 3989	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
4		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	sseldd 3944	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
7	6	rpred 12971	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
8	5, 7	resubcld 11582	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
9	5, 7	readdcld 11179	. . 3 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
10	5, 6	ltsubrpd 13003	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
11	5, 6	ltaddrpd 13004	. . . . 5 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
12	8, 5, 9, 10, 11	lttrd 11311	. . . 4 ⊢ (𝜑 → (𝐶 − 𝑅) < (𝐶 + 𝑅))
13	8, 9, 12	ltled 11298	. . 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 24823	. . . 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 25394	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
19		cncff 24819	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
20	15, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
21	20, 4	ffvelcdmd 7039	. . . . . . 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 11200	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ^*)
24	9	rexrd 11200	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ^*)
25		lbicc2 13401	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 − 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
26	23, 24, 13, 25	syl3anc 1373	. . . . . . . . . . 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 11298	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
29	5, 9, 11	ltled 11298	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
30		elicc2 13348	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
32	5, 28, 29, 31	mpbir3and 1343	. . . . . . . . . . 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 7283	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 6860	. . . . . . . . . 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 6860	. . . . . . . . . 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 5115	. . . . . . . . 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 239	. . . . . . . 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 6674	. . . . . . . . . . . . . 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 6680	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → dom 𝐹 = 𝑋)
48	46, 47	sseqtrrd 3981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
49		funfvima2 7187	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 5644	. . . . . . . . . . . . 13 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
53		simprr 772	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 2830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦))
56		elicc2 13348	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 232	. . . . . . . . . 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 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 − 𝑅)))
60		simprll 778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ)
61	14, 26	sseldd 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ 𝑋)
62	20, 61	ffvelcdmd 7039	. . . . . . . . . . 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 11240	. . . . . . . . . 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 1373	. . . . . . . . 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 695	. . . . . . . 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 13402	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ^* ∧ (𝐶 + 𝑅) ∈ ℝ^* ∧ (𝐶 − 𝑅) ≤ (𝐶 + 𝑅)) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
69	23, 24, 13, 68	syl3anc 1373	. . . . . . . . . . 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 7283	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 6853	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ∈ V
78		fvex 6853	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ∈ V
79	77, 78	brcnv 5836	. . . . . . . . . 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 6860	. . . . . . . . . . 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 5115	. . . . . . . . . 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 239	. . . . . . . 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 7187	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦))
88		elicc2 13348	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 232	. . . . . . . . . 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 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ≤ (𝐹‘(𝐶 + 𝑅)))
92	14, 69	sseldd 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ 𝑋)
93	20, 92	ffvelcdmd 7039	. . . . . . . . . . 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 11240	. . . . . . . . . 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 1373	. . . . . . . . 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 695	. . . . . . . 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 11101	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
100	99	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		fss 6686	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
102	20, 99, 101	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
103	14, 3	sstrd 3954	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
104		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
105		tgioo4 24726	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106	104, 105	dvres 25845	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
108		iccntr 24743	. . . . . . . . . . . . . 14 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
109	8, 9, 108	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
110	109	reseq2d 5939	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
111	107, 110	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
112	111	dmeqd 5859	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
113		dmres 5972	. . . . . . . . . . 11 ⊢ dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹))
114		ioossicc 13370	. . . . . . . . . . . . . 14 ⊢ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))
115	114, 14	sstrid 3955	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ 𝑋)
116	115, 1	sseqtrrd 3981	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹))
117		dfss2 3929	. . . . . . . . . . . 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 218	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹)) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
119	113, 118	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
120	112, 119	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
121		resss 5961	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) ⊆ (ℝ D 𝐹)
122	111, 121	eqsstrdi 3988	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹))
123		rnss 5892	. . . . . . . . . . 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 3946	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
127	8, 9, 17, 120, 126	dvne0 25949	. . . . . . . 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 860	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 < (𝐹‘𝐶))
130		isorel 7283	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 5115	. . . . . . . . 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 239	. . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ≤ 𝑦)
138		simprlr 779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ)
139		ltletr 11242	. . . . . . . . . 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 1373	. . . . . . . . 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 694	. . . . . . . 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 7283	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 6853	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ∈ V
149	148, 77	brcnv 5836	. . . . . . . . . 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 5115	. . . . . . . . . 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 239	. . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ≤ 𝑦)
154		ltletr 11242	. . . . . . . . . 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 1373	. . . . . . . . 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 694	. . . . . . . 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 860	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) < 𝑦)
159	60	rexrd 11200	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ^*)
160	138	rexrd 11200	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ^*)
161		elioo2 13323	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
162	159, 160, 161	syl2anc 584	. . . . . 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 1343	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ (𝑥(,)𝑦))
164	54	fveq2d 6844	. . . . . 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 24743	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
166	165	ad2antrl 728	. . . . . 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 2764	. . . . 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 2831	. . . 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 3192	. 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5102 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6493 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 Isom wiso 6500 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 TopOpenctopn 17360 topGenctg 17376 ℂ_fldccnfld 21296 intcnt 22937 –cn→ccncf 24802 D cdv 25797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: dvcnvrelem2 25956

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