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Theorem dvcnvre 25760

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 2732	. 2 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	tgioo2 24539	. 2 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
3		reelprrecn 11204	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
4	3	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
5		retop 24498	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
6		dvcnvre.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
7		f1ofo 6840	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
8		forn 6808	. . . . . . 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 24633	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
12		frn 6724	. . . . . . 7 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
13	10, 11, 12	3syl 18	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
14	9, 13	eqsstrrd 4021	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
15		uniretop 24499	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	15	ntrss2 22781	. . . . 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 7277	. . . . . 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 2732	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
21	20	rexmet 24527	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
22		dvcnvre.d	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
23		dvbsss 25643	. . . . . . . . . . . . 13 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
25	22, 24	eqsstrrd 4021	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
26	15	ntrss2 22781	. . . . . . . . . . 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 11169	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ⊆ ℂ)
30	10, 11	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
31		fss 6734	. . . . . . . . . . . . 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 25641	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
34	22, 33	eqsstrrd 4021	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ((int‘(topGen‘ran (,)))‘𝑋))
35	27, 34	eqssd 3999	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑋) = 𝑋)
36	15	isopn3 22790	. . . . . . . . . 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 256	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (topGen‘ran (,)))
39		f1ocnv 6845	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
40		f1of 6833	. . . . . . . . . 10 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
41	6, 39, 40	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
42	41	ffvelcdmda 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
43		eqid 2732	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
44	20, 43	tgioo 24532	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
45	44	mopni2 24222	. . . . . . . 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 1468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑟 ∈ ℝ⁺ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)
47	10	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝐹 ∈ (𝑋–cn→ℝ))
48	22	ad2antrr 724	. . . . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐹))
51	6	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝐹:𝑋–1-1-onto→𝑌)
52	42	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (^◡𝐹‘𝑥) ∈ 𝑋)
53		rphalfcl 13005	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
54	53	ad2antrl 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ⁺)
55	25	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑋 ⊆ ℝ)
56	55, 52	sseldd 3983	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (^◡𝐹‘𝑥) ∈ ℝ)
57	54	rpred 13020	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → (𝑟 / 2) ∈ ℝ)
58	56, 57	resubcld 11646	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) − (𝑟 / 2)) ∈ ℝ)
59	56, 57	readdcld 11247	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → ((^◡𝐹‘𝑥) + (𝑟 / 2)) ∈ ℝ)
60		elicc2 13393	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . 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 13020	. . . . . . . . . . . . . . 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 11646	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 13020	. . . . . . . . . . . . . . 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 13007	. . . . . . . . . . . . . . . 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 11831	. . . . . . . . . . . . . 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 11378	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 11247	. . . . . . . . . . . . . 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 11377	. . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . 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 11268	. . . . . . . . . . . . . 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 11268	. . . . . . . . . . . . . 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 13369	. . . . . . . . . . . . . 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 1342	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 3988	. . . . . . . . . 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 12986	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
89	88	ad2antrl 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ (𝑟 ∈ ℝ⁺ ∧ ((^◡𝐹‘𝑥)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝑋)) → 𝑟 ∈ ℝ)
90	20	bl2ioo 24528	. . . . . . . . . . 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 4023	. . . . . . . . 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 771	. . . . . . . . 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 3992	. . . . . . . 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 2732	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		eqid 2732	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑋) = ((TopOpen‘ℂ_fld) ↾_t 𝑋)
97		eqid 2732	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑌) = ((TopOpen‘ℂ_fld) ↾_t 𝑌)
98	47, 48, 50, 51, 52, 54, 94, 95, 1, 96, 97	dvcnvrelem2 25759	. . . . . . 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 3161	. . . . . 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 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) ∈ ((int‘(topGen‘ran (,)))‘𝑌))
101	19, 100	eqeltrrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ ((int‘(topGen‘ran (,)))‘𝑌))
102	17, 101	eqelssd 4003	. . 3 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝑌) = 𝑌)
103	15	isopn3 22790	. . . 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 256	. 2 ⊢ (𝜑 → 𝑌 ∈ (topGen‘ran (,)))
106	99	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘(𝐹‘(^◡𝐹‘𝑥))))
107	19	fveq2d 6895	. . . . . 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 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
109	108	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ^◡𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑌) CnP ((TopOpen‘ℂ_fld) ↾_t 𝑋))‘𝑥))
110	1	cnfldtopon 24519	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
111	14, 28	sstrdi 3994	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
112		resttopon 22885	. . . . . 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 3994	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
115		resttopon 22885	. . . . . 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 23004	. . . . 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 711	. . 3 ⊢ (𝜑 → ^◡𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑌) Cn ((TopOpen‘ℂ_fld) ↾_t 𝑋)))
120	1, 97, 96	cncfcn 24650	. . . 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 2836	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
123	1, 2, 4, 105, 6, 122, 22, 49	dvcnv 25718	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⊆ wss 3948 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 2c2 12271 ℝ⁺crp 12978 (,)cioo 13328 [,]cicc 13331 abscabs 15185 ↾_t crest 17370 TopOpenctopn 17371 topGenctg 17387 ∞Metcxmet 21129 ballcbl 21131 MetOpencmopn 21134 ℂ_fldccnfld 21144 Topctop 22615 TopOnctopon 22632 intcnt 22741 Cn ccn 22948 CnP ccnp 22949 –cn→ccncf 24616 D cdv 25604

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25607 df-dv 25608

This theorem is referenced by: dvrelog 26369

W3C validator