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Theorem dvcnvrelem1 25968

Description: Lemma for dvcnvre 25970. (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 25849	. . . . . 6 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
3	1, 2	eqsstrrdi 4028	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
4		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	sseldd 3973	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
7	6	rpred 13048	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
8	5, 7	resubcld 11672	. . 3 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
9	5, 7	readdcld 11273	. . 3 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
10	5, 6	ltsubrpd 13080	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
11	5, 6	ltaddrpd 13081	. . . . 5 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
12	8, 5, 9, 10, 11	lttrd 11405	. . . 4 ⊢ (𝜑 → (𝐶 − 𝑅) < (𝐶 + 𝑅))
13	8, 9, 12	ltled 11392	. . 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 24835	. . . 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 25407	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))
19		cncff 24831	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
20	15, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
21	20, 4	ffvelcdmd 7090	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
22	21	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) ∈ ℝ)
23	8	rexrd 11294	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ^*)
24	9	rexrd 11294	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ^*)
25		lbicc2 13473	. . . . . . . . . . . 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 479	. . . . . . . . . 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 11392	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
29	5, 9, 11	ltled 11392	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
30		elicc2 13421	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
34	10	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 − 𝑅) < 𝐶)
35		isorel 7330	. . . . . . . . . . . . 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 419	. . . . . . . . . . 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 6912	. . . . . . . . . 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 6912	. . . . . . . . . 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 5156	. . . . . . . . 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 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐹:𝑋⟶ℝ)
45	44	ffund 6721	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → Fun 𝐹)
46	14	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
47	44	fdmd 6728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → dom 𝐹 = 𝑋)
48	46, 47	sseqtrrd 4014	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
49		funfvima2 7239	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 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 5685	. . . . . . . . . . . . 13 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
53		simprr 771	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 2827	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ (𝑥[,]𝑦))
56		elicc2 13421	. . . . . . . . . . . 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 726	. . . . . . . . . . 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 777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ)
61	14, 26	sseldd 3973	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ 𝑋)
62	20, 61	ffvelcdmd 7090	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
63	62	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 − 𝑅)) ∈ ℝ)
64		lelttr 11334	. . . . . . . . . 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 693	. . . . . . . 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 13474	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐶 + 𝑅) ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
71	11	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝐶 < (𝐶 + 𝑅))
72		isorel 7330	. . . . . . . . . . . . 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 419	. . . . . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘𝐶) ∈ V
78		fvex 6905	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 + 𝑅)) ∈ V
79	77, 78	brcnv 5879	. . . . . . . . . 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 6912	. . . . . . . . . . 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 5156	. . . . . . . . . 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 282	. . . . . . . . 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 7239	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 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 2827	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ (𝑥[,]𝑦))
88		elicc2 13421	. . . . . . . . . . . 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 726	. . . . . . . . . . 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 3973	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ 𝑋)
93	20, 92	ffvelcdmd 7090	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
94	93	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘(𝐶 + 𝑅)) ∈ ℝ)
95		lelttr 11334	. . . . . . . . . 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 693	. . . . . . . 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 11195	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
100	99	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
101		fss 6734	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
102	20, 99, 101	sylancl 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
103	14, 3	sstrd 3983	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
104		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
105	104	tgioo2 24737	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106	104, 105	dvres 25858	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
108		iccntr 24755	. . . . . . . . . . . . . 14 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
109	8, 9, 108	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
110	109	reseq2d 5979	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
111	107, 110	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
112	111	dmeqd 5902	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))))
113		dmres 6011	. . . . . . . . . . 11 ⊢ dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = (((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ∩ dom (ℝ D 𝐹))
114		ioossicc 13442	. . . . . . . . . . . . . 14 ⊢ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))
115	114, 14	sstrid 3984	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ 𝑋)
116	115, 1	sseqtrrd 4014	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)) ⊆ dom (ℝ D 𝐹))
117		dfss2 3957	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
120	112, 119	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐶 − 𝑅)(,)(𝐶 + 𝑅)))
121		resss 6001	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ↾ ((𝐶 − 𝑅)(,)(𝐶 + 𝑅))) ⊆ (ℝ D 𝐹)
122	111, 121	eqsstrdi 4027	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ (ℝ D 𝐹))
123		rnss 5935	. . . . . . . . . . 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 3975	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
127	8, 9, 17, 120, 126	dvne0 25962	. . . . . . . 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 479	. . . . . . 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 858	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 < (𝐹‘𝐶))
130		isorel 7330	. . . . . . . . . . . . 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 419	. . . . . . . . . . 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 5156	. . . . . . . . 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 778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ)
139		ltletr 11336	. . . . . . . . . 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 692	. . . . . . . 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 7330	. . . . . . . . . . . . 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 419	. . . . . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))‘(𝐶 − 𝑅)) ∈ V
149	148, 77	brcnv 5879	. . . . . . . . . 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 5156	. . . . . . . . . 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 282	. . . . . . . . 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 11336	. . . . . . . . . 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 692	. . . . . . . 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 858	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → (𝐹‘𝐶) < 𝑦)
159	60	rexrd 11294	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑥 ∈ ℝ^*)
160	138	rexrd 11294	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ran (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑥[,]𝑦))) → 𝑦 ∈ ℝ^*)
161		elioo2 13397	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐶) ∈ (𝑥(,)𝑦) ↔ ((𝐹‘𝐶) ∈ ℝ ∧ 𝑥 < (𝐹‘𝐶) ∧ (𝐹‘𝐶) < 𝑦)))
162	159, 160, 161	syl2anc 582	. . . . . 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 6896	. . . . . 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 24755	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
166	165	ad2antrl 726	. . . . . 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 2765	. . . . 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 2828	. . . 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 455	. . 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 3202	. 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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ∩ cin 3938 ⊆ wss 3939 class class class wbr 5143 ^◡ccnv 5671 dom cdm 5672 ran crn 5673 ↾ cres 5674 “ cima 5675 Fun wfun 6537 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7416 ℂcc 11136 ℝcr 11137 0cc0 11138 + caddc 11141 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 − cmin 11474 ℝ⁺crp 13006 (,)cioo 13356 [,]cicc 13359 TopOpenctopn 17402 topGenctg 17418 ℂ_fldccnfld 21283 intcnt 22939 –cn→ccncf 24814 D cdv 25810

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-cmp 23309 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814

This theorem is referenced by: dvcnvrelem2 25969

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