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Theorem dvcnvlem 26028

Description: Lemma for dvcnvre 26072. (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 6848	. . . . 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 7104	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
6		dvcnv.k	. . . . . 6 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
7		dvcnv.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopon 24818	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
9		dvcnv.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
10		recnprss 25953	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12		resttopon 23184	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
13	8, 11, 12	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
14	6, 13	eqeltrid 2842	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
15		topontop 22934	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top)
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		dvcnv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
18		isopn3i 23105	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌)
19	16, 17, 18	syl2anc 584	. . 3 ⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌)
20	5, 19	eleqtrrd 2841	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌))
21		f1ocnv 6860	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
22		f1of 6848	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
23	1, 21, 22	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
24		eldifi 4140	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌)
25		ffvelcdm 7100	. . . . . . . 8 ⊢ ((^◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (^◡𝐹‘𝑧) ∈ 𝑋)
26	23, 24, 25	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (^◡𝐹‘𝑧) ∈ 𝑋)
27	26	anim1i 615	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
28		eldifsn 4790	. . . . . 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 4140	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋)
32		dvcnv.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
33		dvbsss 25951	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
34	32, 33	eqsstrrdi 4050	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
35	34, 11	sstrd 4005	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
36	35	sselda 3994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ)
37	31, 36	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ)
38	34, 4	sseldd 3995	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
39	11, 38	sseldd 3995	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ)
41	37, 40	subcld 11617	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ)
42		toponss 22948	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
43	14, 17, 42	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
44	43, 11	sstrd 4005	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
45	3, 44	fssd 6753	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
46		ffvelcdm 7100	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ)
47	45, 31, 46	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ)
48	44, 5	sseldd 3995	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ)
50	47, 49	subcld 11617	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ)
51		eldifsni 4794	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶)
52	51	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶)
53	47, 49	subeq0ad 11627	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶)))
54		f1of1 6847	. . . . . . . . . . 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 7281	. . . . . . . . 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 2982	. . . . . 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 12040	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ)
65		limcresi 25934	. . . . . 6 ⊢ (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶))
66	23	feqmptd 6976	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹 = (𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)))
67	66	reseq1d 5998	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})))
68		difss 4145	. . . . . . . . 9 ⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌
69		resmpt 6056	. . . . . . . . 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 2790	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)))
72	71	oveq1d 7445	. . . . . 6 ⊢ (𝜑 → ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
73	65, 72	sseqtrid 4047	. . . . 5 ⊢ (𝜑 → (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
74		f1ocnvfv1 7295	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
75	1, 4, 74	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
76		dvcnv.i	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
77	76, 5	cnlimci 25938	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
78	75, 77	eqeltrrd 2839	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
79	73, 78	sseldd 3995	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
80	45, 35, 4	dvlem 25945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ)
81	37, 40, 52	subne0d 11626	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0)
82	50, 41, 63, 81	divne0d 12056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0)
83		eldifsn 4790	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}))
85	84	fmpttd 7134	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖ {0}))
86		difss 4145	. . . . . . 7 ⊢ (ℂ ∖ {0}) ⊆ ℂ
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
88		eqid 2734	. . . . . 6 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (𝐽 ↾_t (ℂ ∖ {0}))
89	4, 32	eleqtrrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
90		dvfg 25955	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
91		ffun 6739	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
92		funfvbrb 7070	. . . . . . . . . 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 2734	. . . . . . . . 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 25947	. . . . . . . 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 23184	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
100	8, 86, 99	mp2an 692	. . . . . . . . 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 11253	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
104	101, 102, 103	cnmptc 23685	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
105	101	cnmptid 23684	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn (𝐽 ↾_t (ℂ ∖ {0}))))
106	7, 88	divcn 24905	. . . . . . . . 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 23689	. . . . . . 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 6722	. . . . . . . . . 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 7104	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ)
113	109	ffnd 6737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
114		fnfvelrn 7099	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
116		dvcnv.z	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
117		nelne2 3037	. . . . . . . . 9 ⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
118	115, 116, 117	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
119		eldifsn 4790	. . . . . . . 8 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔ (((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0))
120	112, 118, 119	sylanbrc 583	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}))
121	100	toponunii 22937	. . . . . . . 8 ⊢ (ℂ ∖ {0}) = ∪ (𝐽 ↾_t (ℂ ∖ {0}))
122	121	cncnpi 23301	. . . . . . 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 584	. . . . . 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 25940	. . . . 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 7438	. . . . . . 7 ⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶)))
126		eqid 2734	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))
127		ovex 7463	. . . . . . 7 ⊢ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ V
128	125, 126, 127	fvmpt 7015	. . . . . 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 2735	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
131		eqidd 2735	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)))
132		oveq2 7438	. . . . . . . 8 ⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
133	84, 130, 131, 132	fmptco 7148	. . . . . . 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 12057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))
135	134	mpteq2dva 5247	. . . . . . 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 2774	. . . . . 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 7445	. . . . 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 2852	. . . 4 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) lim_ℂ 𝐶))
139		oveq1 7437	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((^◡𝐹‘𝑧) − 𝐶))
140		fveq2 6906	. . . . . 6 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑧)))
141	140	oveq1d 7445	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))
142	139, 141	oveq12d 7448	. . . 4 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))))
143		eldifsni 4794	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶))
144	143	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶))
145	144	necomd 2993	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧)
146		f1ocnvfvb 7298	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
147	1, 4, 24, 146	syl2an3an 1421	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
148	147	necon3abid 2974	. . . . . . 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 25942	. . 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 2740	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
154	153	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
155	154	oveq2d 7446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) − 𝐶) = ((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))))
156		f1ocnvfv2 7296	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
157	1, 24, 156	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
158	157	oveq1d 7445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶)))
159	155, 158	oveq12d 7448	. . . . 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 5247	. . . 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 7445	. . 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 2840	. 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 2734	. . 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 6753	. . 3 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
165	6, 7, 163, 11, 164, 43	eldv 25947	. 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 713	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 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5687 dom cdm 5688 ran crn 5689 ↾ cres 5690 ∘ ccom 5692 Fun wfun 6556 Fn wfn 6557 ⟶wf 6558 –1-1→wf1 6559 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 − cmin 11489 / cdiv 11917 ↾_t crest 17466 TopOpenctopn 17467 ℂ_fldccnfld 21381 Topctop 22914 TopOnctopon 22931 intcnt 23040 Cn ccn 23247 CnP ccnp 23248 ×_t ctx 23583 –cn→ccncf 24915 lim_ℂ climc 25911 D cdv 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916

This theorem is referenced by: dvcnv 26029

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