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Theorem dvcnvlem 25045

Description: Lemma for dvcnvre 25088. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
dvcnv.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvcnv.k	⊢ 𝐾 = (𝐽 ↾_t 𝑆)
dvcnv.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvcnv.y	⊢ (𝜑 → 𝑌 ∈ 𝐾)
dvcnv.f	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnv.i	⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
dvcnv.d	⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
dvcnv.z	⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
dvcnv.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)

**Assertion**
Ref	Expression
dvcnvlem	⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)))

Proof of Theorem dvcnvlem Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcnv.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
2		f1of 6700	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
4		dvcnv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5	3, 4	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
6		dvcnv.k	. . . . . 6 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
7		dvcnv.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopon 23852	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
9		dvcnv.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
10		recnprss 24973	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12		resttopon 22220	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
13	8, 11, 12	sylancr 586	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
14	6, 13	eqeltrid 2843	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
15		topontop 21970	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top)
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		dvcnv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
18		isopn3i 22141	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌)
19	16, 17, 18	syl2anc 583	. . 3 ⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌)
20	5, 19	eleqtrrd 2842	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌))
21		f1ocnv 6712	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
22		f1of 6700	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
23	1, 21, 22	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
24		eldifi 4057	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌)
25		ffvelrn 6941	. . . . . . . 8 ⊢ ((^◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (^◡𝐹‘𝑧) ∈ 𝑋)
26	23, 24, 25	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (^◡𝐹‘𝑧) ∈ 𝑋)
27	26	anim1i 614	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
28		eldifsn 4717	. . . . . 6 ⊢ ((^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}) ↔ ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
29	27, 28	sylibr 233	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → (^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}))
30	29	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (^◡𝐹‘𝑧) ≠ 𝐶)) → (^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}))
31		eldifi 4057	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋)
32		dvcnv.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
33		dvbsss 24971	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
34	32, 33	eqsstrrdi 3972	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
35	34, 11	sstrd 3927	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
36	35	sselda 3917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ)
37	31, 36	sylan2 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ)
38	34, 4	sseldd 3918	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
39	11, 38	sseldd 3918	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ)
41	37, 40	subcld 11262	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ)
42		toponss 21984	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
43	14, 17, 42	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
44	43, 11	sstrd 3927	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
45	3, 44	fssd 6602	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
46		ffvelrn 6941	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ)
47	45, 31, 46	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ)
48	44, 5	sseldd 3918	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ)
50	47, 49	subcld 11262	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ)
51		eldifsni 4720	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶)
52	51	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶)
53	47, 49	subeq0ad 11272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶)))
54		f1of1 6699	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
55	1, 54	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)
56	55	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐹:𝑋–1-1→𝑌)
57	31	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ 𝑋)
58	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ 𝑋)
59		f1fveq 7116	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑦 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
60	56, 57, 58, 59	syl12anc 833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
61	53, 60	bitrd 278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ 𝑦 = 𝐶))
62	61	necon3bid 2987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0 ↔ 𝑦 ≠ 𝐶))
63	52, 62	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0)
64	41, 50, 63	divcld 11681	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ)
65		limcresi 24954	. . . . . 6 ⊢ (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶))
66	23	feqmptd 6819	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹 = (𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)))
67	66	reseq1d 5879	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})))
68		difss 4062	. . . . . . . . 9 ⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌
69		resmpt 5934	. . . . . . . . 9 ⊢ ((𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)))
70	68, 69	ax-mp 5	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧))
71	67, 70	eqtrdi 2795	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)))
72	71	oveq1d 7270	. . . . . 6 ⊢ (𝜑 → ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
73	65, 72	sseqtrid 3969	. . . . 5 ⊢ (𝜑 → (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
74		f1ocnvfv1 7129	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
75	1, 4, 74	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
76		dvcnv.i	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
77	76, 5	cnlimci 24958	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
78	75, 77	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
79	73, 78	sseldd 3918	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
80	45, 35, 4	dvlem 24965	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ)
81	37, 40, 52	subne0d 11271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0)
82	50, 41, 63, 81	divne0d 11697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0)
83		eldifsn 4717	. . . . . . . 8 ⊢ ((((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}) ↔ ((((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ ∧ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0))
84	80, 82, 83	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}))
85	84	fmpttd 6971	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖ {0}))
86		difss 4062	. . . . . . 7 ⊢ (ℂ ∖ {0}) ⊆ ℂ
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
88		eqid 2738	. . . . . 6 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (𝐽 ↾_t (ℂ ∖ {0}))
89	4, 32	eleqtrrd 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
90		dvfg 24975	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
91		ffun 6587	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
92		funfvbrb 6910	. . . . . . . . . 10 ⊢ (Fun (𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
93	9, 90, 91, 92	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
94	89, 93	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))
95		eqid 2738	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))
96	6, 7, 95, 11, 45, 34	eldv 24967	. . . . . . . 8 ⊢ (𝜑 → (𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶) ↔ (𝐶 ∈ ((int‘𝐾)‘𝑋) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) lim_ℂ 𝐶))))
97	94, 96	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ ((int‘𝐾)‘𝑋) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) lim_ℂ 𝐶)))
98	97	simprd 495	. . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) lim_ℂ 𝐶))
99		resttopon 22220	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
100	8, 86, 99	mp2an 688	. . . . . . . . 9 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0}))
101	100	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
102	8	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
103		1cnd 10901	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
104	101, 102, 103	cnmptc 22721	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
105	101	cnmptid 22720	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn (𝐽 ↾_t (ℂ ∖ {0}))))
106	7, 88	divcn 23937	. . . . . . . . 9 ⊢ / ∈ ((𝐽 ×_t (𝐽 ↾_t (ℂ ∖ {0}))) Cn 𝐽)
107	106	a1i 11	. . . . . . . 8 ⊢ (𝜑 → / ∈ ((𝐽 ×_t (𝐽 ↾_t (ℂ ∖ {0}))) Cn 𝐽))
108	101, 104, 105, 107	cnmpt12f 22725	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
109	9, 90	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
110	32	feq2d 6570	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ))
111	109, 110	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ)
112	111, 4	ffvelrnd 6944	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ)
113	109	ffnd 6585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
114		fnfvelrn 6940	. . . . . . . . . 10 ⊢ (((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ 𝐶 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
115	113, 89, 114	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
116		dvcnv.z	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
117		nelne2 3041	. . . . . . . . 9 ⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
118	115, 116, 117	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
119		eldifsn 4717	. . . . . . . 8 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔ (((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0))
120	112, 118, 119	sylanbrc 582	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}))
121	100	toponunii 21973	. . . . . . . 8 ⊢ (ℂ ∖ {0}) = ∪ (𝐽 ↾_t (ℂ ∖ {0}))
122	121	cncnpi 22337	. . . . . . 7 ⊢ (((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0})) → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ (((𝐽 ↾_t (ℂ ∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶)))
123	108, 120, 122	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ (((𝐽 ↾_t (ℂ ∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶)))
124	85, 87, 7, 88, 98, 123	limccnp 24960	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) ∈ (((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) lim_ℂ 𝐶))
125		oveq2 7263	. . . . . . 7 ⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶)))
126		eqid 2738	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))
127		ovex 7288	. . . . . . 7 ⊢ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ V
128	125, 126, 127	fvmpt 6857	. . . . . 6 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶)))
129	120, 128	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶)))
130		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
131		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)))
132		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
133	84, 130, 131, 132	fmptco 6983	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))))
134	50, 41, 63, 81	recdivd 11698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))
135	134	mpteq2dva 5170	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))))
136	133, 135	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))))
137	136	oveq1d 7270	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) lim_ℂ 𝐶) = ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) lim_ℂ 𝐶))
138	124, 129, 137	3eltr3d 2853	. . . 4 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) lim_ℂ 𝐶))
139		oveq1 7262	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((^◡𝐹‘𝑧) − 𝐶))
140		fveq2 6756	. . . . . 6 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑧)))
141	140	oveq1d 7270	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))
142	139, 141	oveq12d 7273	. . . 4 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))))
143		eldifsni 4720	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶))
144	143	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶))
145	144	necomd 2998	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧)
146		f1ocnvfvb 7132	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
147	1, 4, 24, 146	syl2an3an 1420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
148	147	necon3abid 2979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) ≠ 𝑧 ↔ ¬ (^◡𝐹‘𝑧) = 𝐶))
149	145, 148	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ¬ (^◡𝐹‘𝑧) = 𝐶)
150	149	pm2.21d 121	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) = 𝐶 → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶))))
151	150	impr 454	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (^◡𝐹‘𝑧) = 𝐶)) → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶)))
152	30, 64, 79, 138, 142, 151	limcco 24962	. . 3 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))
153	75	eqcomd 2744	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
154	153	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
155	154	oveq2d 7271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) − 𝐶) = ((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))))
156		f1ocnvfv2 7130	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
157	1, 24, 156	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
158	157	oveq1d 7270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶)))
159	155, 158	oveq12d 7273	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶))))
160	159	mpteq2dva 5170	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))))
161	160	oveq1d 7270	. . 3 ⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))
162	152, 161	eleqtrd 2841	. 2 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))
163		eqid 2738	. . 3 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶))))
164	23, 35	fssd 6602	. . 3 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
165	6, 7, 163, 11, 164, 43	eldv 24967	. 2 ⊢ (𝜑 → ((𝐹‘𝐶)(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)) ↔ ((𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌) ∧ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))))
166	20, 162, 165	mpbir2and 709	1 ⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 − cmin 11135 / cdiv 11562 ↾_t crest 17048 TopOpenctopn 17049 ℂ_fldccnfld 20510 Topctop 21950 TopOnctopon 21967 intcnt 22076 Cn ccn 22283 CnP ccnp 22284 ×_t ctx 22619 –cn→ccncf 23945 lim_ℂ climc 24931 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: dvcnv 25046

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