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Theorem dvcnvre 25996

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 2737	. 2 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2		tgioo4 24780	. 2 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
3		reelprrecn 11121	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
5		retop 24736	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
6		dvcnvre.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
7		f1ofo 6781	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8		forn 6749	. . . . . . 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 24870	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
12		frn 6669	. . . . . . 7 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
13	10, 11, 12	3syl 18	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
14	9, 13	eqsstrrd 3958	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
15		uniretop 24737	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	15	ntrss2 23032	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑌 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝑌) ⊆ 𝑌)
17	5, 14, 16	sylancr 588	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑌) ⊆ 𝑌)
18		f1ocnvfv2 7225	. . . . . 6 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
19	6, 18	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
20		eqid 2737	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
21	20	rexmet 24766	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
22		dvcnvre.d	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
23		dvbsss 25879	. . . . . . . . . . . . 13 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
25	22, 24	eqsstrrd 3958	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
26	15	ntrss2 23032	. . . . . . . . . . 11 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑋 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝑋) ⊆ 𝑋)
27	5, 25, 26	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑋) ⊆ 𝑋)
28		ax-resscn 11086	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ⊆ ℂ)
30	10, 11	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
31		fss 6678	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝑋⟶ℂ)
32	30, 28, 31	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
33	29, 32, 25, 2, 1	dvbssntr 25877	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
34	22, 33	eqsstrrd 3958	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
35	27, 34	eqssd 3940	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋)
36	15	isopn3 23041	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑋 ⊆ ℝ) → (𝑋 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋))
37	5, 25, 36	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋))
38	35, 37	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (topGen‘ran (,)))
39		f1ocnv 6786	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
40		f1of 6774	. . . . . . . . . 10 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
41	6, 39, 40	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
42	41	ffvelcdmda 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
43		eqid 2737	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
44	20, 43	tgioo 24771	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
45	44	mopni2 24468	. . . . . . . 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 1471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑟 ∈ ℝ⁺ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)
47	10	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝐹 ∈ (𝑋–cn→ℝ))
48	22	ad2antrr 727	. . . . . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐹))
51	6	ad2antrr 727	. . . . . . . 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 12962	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
54	53	ad2antrl 729	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ⁺)
55	25	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑋 ⊆ ℝ)
56	55, 52	sseldd 3923	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (^◡𝐹‘𝑥) ∈ ℝ)
57	54	rpred 12977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ)
58	56, 57	resubcld 11569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) − (𝑟 / 2)) ∈ ℝ)
59	56, 57	readdcld 11165	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) + (𝑟 / 2)) ∈ ℝ)
60		elicc2 13355	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 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 12977	. . . . . . . . . . . . . . 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 11569	. . . . . . . . . . . . . 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 12977	. . . . . . . . . . . . . . 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 12964	. . . . . . . . . . . . . . . 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 11754	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . 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 11297	. . . . . . . . . . . . 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 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)
78	62	simp3d 1145	. . . . . . . . . . . . . 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 11296	. . . . . . . . . . . . . 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 11295	. . . . . . . . . . . . 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 11186	. . . . . . . . . . . . . 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 11186	. . . . . . . . . . . . . 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 13330	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 1344	. . . . . . . . . . . 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 3928	. . . . . . . . . 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 12942	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
89	88	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑟 ∈ ℝ)
90	20	bl2ioo 24767	. . . . . . . . . . 11 ⊢ (((^◡𝐹‘𝑥) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (((^◡𝐹‘𝑥) − 𝑟)(,)((^◡𝐹‘𝑥) + 𝑟)))
91	56, 89, 90	syl2anc 585	. . . . . . . . . 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 3960	. . . . . . . . 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 773	. . . . . . . . 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 3933	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		eqid 2737	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑋) = ((TopOpen‘ℂ_fld) ↾_t 𝑋)
97		eqid 2737	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑌) = ((TopOpen‘ℂ_fld) ↾_t 𝑌)
98	47, 48, 50, 51, 52, 54, 94, 95, 1, 96, 97	dvcnvrelem2 25995	. . . . . . 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 3145	. . . . . 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 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ ((int‘(topGen‘ran (,)))‘𝑌))
102	17, 101	eqelssd 3944	. . 3 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌)
103	15	isopn3 23041	. . . 4 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝑌 ⊆ ℝ) → (𝑌 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌))
104	5, 14, 103	sylancr 588	. . 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 6838	. . . . . 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 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
109	108	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
110	1	cnfldtopon 24757	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	14, 28	sstrdi 3935	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
112		resttopon 23136	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑌 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑌) ∈ (TopOn‘𝑌))
113	110, 111, 112	sylancr 588	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑌) ∈ (TopOn‘𝑌))
114	25, 28	sstrdi 3935	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
115		resttopon 23136	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑋) ∈ (TopOn‘𝑋))
116	110, 114, 115	sylancr 588	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑋) ∈ (TopOn‘𝑋))
117		cncnp 23255	. . . . 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 585	. . . 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 714	. . 3 ⊢ (𝜑 → ^◡𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
120	1, 97, 96	cncfcn 24887	. . . 4 ⊢ ((𝑌 ⊆ ℂ ∧ 𝑋 ⊆ ℂ) → (𝑌–cn→𝑋) = (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
121	111, 114, 120	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑌–cn→𝑋) = (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
122	119, 121	eleqtrrd 2840	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
123	1, 2, 4, 105, 6, 122, 22, 49	dvcnv 25954	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 ∘ ccom 5628 ⟶wf 6488 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 2c2 12227 ℝ⁺crp 12933 (,)cioo 13289 [,]cicc 13292 abscabs 15187 ↾_t crest 17374 TopOpenctopn 17375 topGenctg 17391 ∞Metcxmet 21329 ballcbl 21331 MetOpencmopn 21334 ℂ_fldccnfld 21344 Topctop 22868 TopOnctopon 22885 intcnt 22992 Cn ccn 23199 CnP ccnp 23200 –cn→ccncf 24853 D cdv 25840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844

This theorem is referenced by: dvrelog 26614

W3C validator