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Theorem dvcnvre 25922

Description: The derivative rule for inverse functions. If 𝐹 is a continuous and differentiable bijective function from 𝑋 to 𝑌 which never has derivative 0, then ^◡𝐹 is also differentiable, and its derivative is the reciprocal of the derivative of 𝐹. (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)

**Assertion**
Ref	Expression
dvcnvre	⊢ (𝜑 → (ℝ D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((ℝ D 𝐹)‘(^◡𝐹‘𝑥)))))

Distinct variable groups: 𝑥,𝐹 𝜑,𝑥 𝑥,𝑋 𝑥,𝑌

Proof of Theorem dvcnvre Dummy variables 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2		tgioo4 24691	. 2 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
3		reelprrecn 11101	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
5		retop 24647	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
6		dvcnvre.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
7		f1ofo 6771	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8		forn 6739	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
9	6, 7, 8	3syl 18	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = 𝑌)
10		dvcnvre.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
11		cncff 24784	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
12		frn 6659	. . . . . . 7 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
13	10, 11, 12	3syl 18	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
14	9, 13	eqsstrrd 3971	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
15		uniretop 24648	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	15	ntrss2 22942	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑌 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝑌) ⊆ 𝑌)
17	5, 14, 16	sylancr 587	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑌) ⊆ 𝑌)
18		f1ocnvfv2 7214	. . . . . 6 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
19	6, 18	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
20		eqid 2729	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
21	20	rexmet 24677	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
22		dvcnvre.d	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
23		dvbsss 25801	. . . . . . . . . . . . 13 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
25	22, 24	eqsstrrd 3971	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
26	15	ntrss2 22942	. . . . . . . . . . 11 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑋 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝑋) ⊆ 𝑋)
27	5, 25, 26	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑋) ⊆ 𝑋)
28		ax-resscn 11066	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ⊆ ℂ)
30	10, 11	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
31		fss 6668	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
32	30, 28, 31	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
33	29, 32, 25, 2, 1	dvbssntr 25799	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
34	22, 33	eqsstrrd 3971	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
35	27, 34	eqssd 3953	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋)
36	15	isopn3 22951	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑋 ⊆ ℝ) → (𝑋 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋))
37	5, 25, 36	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋))
38	35, 37	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (topGen‘ran (,)))
39		f1ocnv 6776	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
40		f1of 6764	. . . . . . . . . 10 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
41	6, 39, 40	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
42	41	ffvelcdmda 7018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
43		eqid 2729	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
44	20, 43	tgioo 24682	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
45	44	mopni2 24379	. . . . . . . 8 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝑋 ∈ (topGen‘ran (,)) ∧ (^◡𝐹‘𝑥) ∈ 𝑋) → ∃𝑟 ∈ ℝ⁺ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)
46	21, 38, 42, 45	mp3an2ani 1470	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑟 ∈ ℝ⁺ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)
47	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝐹 ∈ (𝑋–cn→ℝ))
48	22	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → dom (ℝ D 𝐹) = 𝑋)
49		dvcnvre.z	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
50	49	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐹))
51	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝐹:𝑋–1-1-onto→𝑌)
52	42	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (^◡𝐹‘𝑥) ∈ 𝑋)
53		rphalfcl 12922	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
54	53	ad2antrl 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ⁺)
55	25	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑋 ⊆ ℝ)
56	55, 52	sseldd 3936	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (^◡𝐹‘𝑥) ∈ ℝ)
57	54	rpred 12937	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ)
58	56, 57	resubcld 11548	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) − (𝑟 / 2)) ∈ ℝ)
59	56, 57	readdcld 11144	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) + (𝑟 / 2)) ∈ ℝ)
60		elicc2 13314	. . . . . . . . . . . . . . . 16 ⊢ ((((^◡𝐹‘𝑥) − (𝑟 / 2)) ∈ ℝ ∧ ((^◡𝐹‘𝑥) + (𝑟 / 2)) ∈ ℝ) → (𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) ↔ (𝑦 ∈ ℝ ∧ ((^◡𝐹‘𝑥) − (𝑟 / 2)) ≤ 𝑦 ∧ 𝑦 ≤ ((^◡𝐹‘𝑥) + (𝑟 / 2)))))
61	58, 59, 60	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) ↔ (𝑦 ∈ ℝ ∧ ((^◡𝐹‘𝑥) − (𝑟 / 2)) ≤ 𝑦 ∧ 𝑦 ≤ ((^◡𝐹‘𝑥) + (𝑟 / 2)))))
62	61	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (𝑦 ∈ ℝ ∧ ((^◡𝐹‘𝑥) − (𝑟 / 2)) ≤ 𝑦 ∧ 𝑦 ≤ ((^◡𝐹‘𝑥) + (𝑟 / 2))))
63	62	simp1d 1142	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑦 ∈ ℝ)
64	56	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (^◡𝐹‘𝑥) ∈ ℝ)
65		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑟 ∈ ℝ⁺)
66	65	rpred 12937	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑟 ∈ ℝ)
67	64, 66	resubcld 11548	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − 𝑟) ∈ ℝ)
68	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − (𝑟 / 2)) ∈ ℝ)
69	65, 53	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (𝑟 / 2) ∈ ℝ⁺)
70	69	rpred 12937	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (𝑟 / 2) ∈ ℝ)
71		rphalflt 12924	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) < 𝑟)
72	65, 71	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (𝑟 / 2) < 𝑟)
73	70, 66, 64, 72	ltsub2dd 11733	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − 𝑟) < ((^◡𝐹‘𝑥) − (𝑟 / 2)))
74	62	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − (𝑟 / 2)) ≤ 𝑦)
75	67, 68, 63, 73, 74	ltletrd 11276	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − 𝑟) < 𝑦)
76	59	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) + (𝑟 / 2)) ∈ ℝ)
77	64, 66	readdcld 11144	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) + 𝑟) ∈ ℝ)
78	62	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑦 ≤ ((^◡𝐹‘𝑥) + (𝑟 / 2)))
79	70, 66, 64, 72	ltadd2dd 11275	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) + (𝑟 / 2)) < ((^◡𝐹‘𝑥) + 𝑟))
80	63, 76, 77, 78, 79	lelttrd 11274	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑦 < ((^◡𝐹‘𝑥) + 𝑟))
81	67	rexrd 11165	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) − 𝑟) ∈ ℝ^*)
82	77	rexrd 11165	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → ((^◡𝐹‘𝑥) + 𝑟) ∈ ℝ^*)
83		elioo2 13289	. . . . . . . . . . . . . 14 ⊢ ((((^◡𝐹‘𝑥) − 𝑟) ∈ ℝ^* ∧ ((^◡𝐹‘𝑥) + 𝑟) ∈ ℝ^*) → (𝑦 ∈ (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)) ↔ (𝑦 ∈ ℝ ∧ ((^◡𝐹‘𝑥) − 𝑟) < 𝑦 ∧ 𝑦 < ((^◡𝐹‘𝑥) + 𝑟))))
84	81, 82, 83	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → (𝑦 ∈ (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)) ↔ (𝑦 ∈ ℝ ∧ ((^◡𝐹‘𝑥) − 𝑟) < 𝑦 ∧ 𝑦 < ((^◡𝐹‘𝑥) + 𝑟))))
85	63, 75, 80, 84	mpbir3and 1343	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) ∧ 𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2)))) → 𝑦 ∈ (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)))
86	85	ex 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑦 ∈ (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) → 𝑦 ∈ (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟))))
87	86	ssrdv 3941	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) ⊆ (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)))
88		rpre 12902	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
89	88	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑟 ∈ ℝ)
90	20	bl2ioo 24678	. . . . . . . . . . 11 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)))
91	56, 89, 90	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)))
92	87, 91	sseqtrrd 3973	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) ⊆ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟))
93		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)
94	92, 93	sstrd 3946	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (((^◡𝐹‘𝑥) − (𝑟 / 2))[,]((^◡𝐹‘𝑥) + (𝑟 / 2))) ⊆ 𝑋)
95		eqid 2729	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		eqid 2729	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑋) = ((TopOpen‘ℂ_fld) ↾_t 𝑋)
97		eqid 2729	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑌) = ((TopOpen‘ℂ_fld) ↾_t 𝑌)
98	47, 48, 50, 51, 52, 54, 94, 95, 1, 96, 97	dvcnvrelem2 25921	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((𝐹‘(^◡𝐹‘𝑥)) ∈ ((int‘(topGen‘ran (,)))‘𝑌) ∧ ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘(𝐹‘(^◡𝐹‘𝑥)))))
99	46, 98	rexlimddv 3136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(^◡𝐹‘𝑥)) ∈ ((int‘(topGen‘ran (,)))‘𝑌) ∧ ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘(𝐹‘(^◡𝐹‘𝑥)))))
100	99	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) ∈ ((int‘(topGen‘ran (,)))‘𝑌))
101	19, 100	eqeltrrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ ((int‘(topGen‘ran (,)))‘𝑌))
102	17, 101	eqelssd 3957	. . 3 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌)
103	15	isopn3 22951	. . . 4 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑌 ⊆ ℝ) → (𝑌 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌))
104	5, 14, 103	sylancr 587	. . 3 ⊢ (𝜑 → (𝑌 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌))
105	102, 104	mpbird 257	. 2 ⊢ (𝜑 → 𝑌 ∈ (topGen‘ran (,)))
106	99	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘(𝐹‘(^◡𝐹‘𝑥))))
107	19	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘(𝐹‘(^◡𝐹‘𝑥))) = ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
108	106, 107	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
109	108	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
110	1	cnfldtopon 24668	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	14, 28	sstrdi 3948	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
112		resttopon 23046	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑌 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑌) ∈ (TopOn‘𝑌))
113	110, 111, 112	sylancr 587	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑌) ∈ (TopOn‘𝑌))
114	25, 28	sstrdi 3948	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
115		resttopon 23046	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑋) ∈ (TopOn‘𝑋))
116	110, 114, 115	sylancr 587	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑋) ∈ (TopOn‘𝑋))
117		cncnp 23165	. . . . 5 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) ∈ (TopOn‘𝑌) ∧ ((TopOpen‘ℂ_fld) ↾_t 𝑋) ∈ (TopOn‘𝑋)) → (^◡𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)) ↔ (^◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))))
118	113, 116, 117	syl2anc 584	. . . 4 ⊢ (𝜑 → (^◡𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)) ↔ (^◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))))
119	41, 109, 118	mpbir2and 713	. . 3 ⊢ (𝜑 → ^◡𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
120	1, 97, 96	cncfcn 24801	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ⊆ ℂ) → (𝑌–cn→𝑋) = (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
121	111, 114, 120	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑌–cn→𝑋) = (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
122	119, 121	eleqtrrd 2831	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
123	1, 2, 4, 105, 6, 122, 22, 49	dvcnv 25879	1 ⊢ (𝜑 → (ℝ D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((ℝ D 𝐹)‘(^◡𝐹‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ⊆ wss 3903 {cpr 4579 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 ↾ cres 5621 ∘ ccom 5623 ⟶wf 6478 –onto→wfo 6480 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 2c2 12183 ℝ⁺crp 12893 (,)cioo 13248 [,]cicc 13251 abscabs 15141 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ∞Metcxmet 21246 ballcbl 21248 MetOpencmopn 21251 ℂ_fldccnfld 21261 Topctop 22778 TopOnctopon 22795 intcnt 22902 Cn ccn 23109 CnP ccnp 23110 –cn→ccncf 24767 D cdv 25762

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766

This theorem is referenced by: dvrelog 26544

W3C validator