dvcnvlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvcnvlem		Structured version Visualization version GIF version

Theorem dvcnvlem 25948

Description: Lemma for dvcnvre 25992. (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 6782	. . . . 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	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
6		dvcnv.k	. . . . . 6 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
7		dvcnv.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopon 24738	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
9		dvcnv.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
10		recnprss 25873	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12		resttopon 23117	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
13	8, 11, 12	sylancr 588	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
14	6, 13	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
15		topontop 22869	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top)
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		dvcnv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
18		isopn3i 23038	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌)
19	16, 17, 18	syl2anc 585	. . 3 ⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌)
20	5, 19	eleqtrrd 2840	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌))
21		f1ocnv 6794	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
22		f1of 6782	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
23	1, 21, 22	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
24		eldifi 4085	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌)
25		ffvelcdm 7035	. . . . . . . 8 ⊢ ((^◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (^◡𝐹‘𝑧) ∈ 𝑋)
26	23, 24, 25	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (^◡𝐹‘𝑧) ∈ 𝑋)
27	26	anim1i 616	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
28		eldifsn 4744	. . . . . 6 ⊢ ((^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}) ↔ ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
29	27, 28	sylibr 234	. . . . 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 4085	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋)
32		dvcnv.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
33		dvbsss 25871	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
34	32, 33	eqsstrrdi 3981	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
35	34, 11	sstrd 3946	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
36	35	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ)
37	31, 36	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ)
38	34, 4	sseldd 3936	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
39	11, 38	sseldd 3936	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ)
41	37, 40	subcld 11504	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ)
42		toponss 22883	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
43	14, 17, 42	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
44	43, 11	sstrd 3946	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
45	3, 44	fssd 6687	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
46		ffvelcdm 7035	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ)
47	45, 31, 46	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ)
48	44, 5	sseldd 3936	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ)
50	47, 49	subcld 11504	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ)
51		eldifsni 4748	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶)
52	51	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶)
53	47, 49	subeq0ad 11514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶)))
54		f1of1 6781	. . . . . . . . . . 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 7218	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑦 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
60	56, 57, 58, 59	syl12anc 837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
61	53, 60	bitrd 279	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ 𝑦 = 𝐶))
62	61	necon3bid 2977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0 ↔ 𝑦 ≠ 𝐶))
63	52, 62	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0)
64	41, 50, 63	divcld 11929	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ)
65		limcresi 25854	. . . . . 6 ⊢ (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶))
66	23	feqmptd 6910	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹 = (𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)))
67	66	reseq1d 5945	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})))
68		difss 4090	. . . . . . . . 9 ⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌
69		resmpt 6004	. . . . . . . . 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 2788	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)))
72	71	oveq1d 7383	. . . . . 6 ⊢ (𝜑 → ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
73	65, 72	sseqtrid 3978	. . . . 5 ⊢ (𝜑 → (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
74		f1ocnvfv1 7232	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
75	1, 4, 74	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
76		dvcnv.i	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
77	76, 5	cnlimci 25858	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
78	75, 77	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
79	73, 78	sseldd 3936	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
80	45, 35, 4	dvlem 25865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ)
81	37, 40, 52	subne0d 11513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0)
82	50, 41, 63, 81	divne0d 11945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0)
83		eldifsn 4744	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}))
85	84	fmpttd 7069	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖ {0}))
86		difss 4090	. . . . . . 7 ⊢ (ℂ ∖ {0}) ⊆ ℂ
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
88		eqid 2737	. . . . . 6 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (𝐽 ↾_t (ℂ ∖ {0}))
89	4, 32	eleqtrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
90		dvfg 25875	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
91		ffun 6673	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
92		funfvbrb 7005	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))
95		eqid 2737	. . . . . . . . 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 25867	. . . . . . . 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 232	. . . . . . 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 23117	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
100	8, 86, 99	mp2an 693	. . . . . . . . 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 11139	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
104	101, 102, 103	cnmptc 23618	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
105	101	cnmptid 23617	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn (𝐽 ↾_t (ℂ ∖ {0}))))
106	7, 88	divcn 24827	. . . . . . . . 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 23622	. . . . . . 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 6654	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ))
111	109, 110	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ)
112	111, 4	ffvelcdmd 7039	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ)
113	109	ffnd 6671	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
114		fnfvelrn 7034	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
116		dvcnv.z	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
117		nelne2 3031	. . . . . . . . 9 ⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
118	115, 116, 117	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
119		eldifsn 4744	. . . . . . . 8 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔ (((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0))
120	112, 118, 119	sylanbrc 584	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}))
121	100	toponunii 22872	. . . . . . . 8 ⊢ (ℂ ∖ {0}) = ∪ (𝐽 ↾_t (ℂ ∖ {0}))
122	121	cncnpi 23234	. . . . . . 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 585	. . . . . 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 25860	. . . . 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 7376	. . . . . . 7 ⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶)))
126		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))
127		ovex 7401	. . . . . . 7 ⊢ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ V
128	125, 126, 127	fvmpt 6949	. . . . . 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 2738	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
131		eqidd 2738	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)))
132		oveq2 7376	. . . . . . . 8 ⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
133	84, 130, 131, 132	fmptco 7084	. . . . . . 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 11946	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))
135	134	mpteq2dva 5193	. . . . . . 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 2772	. . . . . 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 7383	. . . . 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 2851	. . . 4 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) lim_ℂ 𝐶))
139		oveq1 7375	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((^◡𝐹‘𝑧) − 𝐶))
140		fveq2 6842	. . . . . 6 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑧)))
141	140	oveq1d 7383	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))
142	139, 141	oveq12d 7386	. . . 4 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))))
143		eldifsni 4748	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶))
144	143	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶))
145	144	necomd 2988	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧)
146		f1ocnvfvb 7235	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
147	1, 4, 24, 146	syl2an3an 1425	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
148	147	necon3abid 2969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) ≠ 𝑧 ↔ ¬ (^◡𝐹‘𝑧) = 𝐶))
149	145, 148	mpbid 232	. . . . . 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 25862	. . 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 2743	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
154	153	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
155	154	oveq2d 7384	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) − 𝐶) = ((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))))
156		f1ocnvfv2 7233	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
157	1, 24, 156	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
158	157	oveq1d 7383	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶)))
159	155, 158	oveq12d 7386	. . . . 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 5193	. . . 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 7383	. . 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 2839	. 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 2737	. . 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 6687	. . 3 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
165	6, 7, 163, 11, 164, 43	eldv 25867	. 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 714	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5631 dom cdm 5632 ran crn 5633 ↾ cres 5634 ∘ ccom 5636 Fun wfun 6494 Fn wfn 6495 ⟶wf 6496 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 − cmin 11376 / cdiv 11806 ↾_t crest 17352 TopOpenctopn 17353 ℂ_fldccnfld 21321 Topctop 22849 TopOnctopon 22866 intcnt 22973 Cn ccn 23180 CnP ccnp 23181 ×_t ctx 23516 –cn→ccncf 24837 lim_ℂ climc 25831 D cdv 25832

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: dvcnv 25949

W3C validator