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Theorem dvcnvlem 23543

Description: Lemma for dvcnvre 23586. (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 6050	. . . . 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 6268	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
6		dvcnv.k	. . . . . 6 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
7		dvcnv.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
8	7	cnfldtopon 22396	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
9		dvcnv.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
10		recnprss 23474	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12		resttopon 20775	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
13	8, 11, 12	sylancr 694	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
14	6, 13	syl5eqel 2692	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
15		topontop 20541	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top)
16	14, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
17		dvcnv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
18		isopn3i 20696	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌)
19	16, 17, 18	syl2anc 691	. . 3 ⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌)
20	5, 19	eleqtrrd 2691	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌))
21		f1ocnv 6062	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
22		f1of 6050	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
23	1, 21, 22	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
24		eldifi 3694	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌)
25		ffvelrn 6265	. . . . . . . 8 ⊢ ((^◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (^◡𝐹‘𝑧) ∈ 𝑋)
26	23, 24, 25	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (^◡𝐹‘𝑧) ∈ 𝑋)
27	26	anim1i 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
28		eldifsn 4260	. . . . . 6 ⊢ ((^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}) ↔ ((^◡𝐹‘𝑧) ∈ 𝑋 ∧ (^◡𝐹‘𝑧) ≠ 𝐶))
29	27, 28	sylibr 223	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (^◡𝐹‘𝑧) ≠ 𝐶) → (^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}))
30	29	anasss 677	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (^◡𝐹‘𝑧) ≠ 𝐶)) → (^◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}))
31		eldifi 3694	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋)
32		dvcnv.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
33		dvbsss 23472	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
34	32, 33	syl6eqssr 3619	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
35	34, 11	sstrd 3578	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
36	35	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ)
37	31, 36	sylan2 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ)
38	34, 4	sseldd 3569	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
39	11, 38	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ)
41	37, 40	subcld 10271	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ)
42		toponss 20544	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
43	14, 17, 42	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
44	43, 11	sstrd 3578	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
45	3, 44	fssd 5970	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
46		ffvelrn 6265	. . . . . . 7 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ)
47	45, 31, 46	syl2an 493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ)
48	44, 5	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ)
50	47, 49	subcld 10271	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ)
51		eldifsni 4261	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶)
52	51	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶)
53	47, 49	subeq0ad 10281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶)))
54		f1of1 6049	. . . . . . . . . . 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 6420	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑦 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
60	56, 57, 58, 59	syl12anc 1316	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶))
61	53, 60	bitrd 267	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ 𝑦 = 𝐶))
62	61	necon3bid 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0 ↔ 𝑦 ≠ 𝐶))
63	52, 62	mpbird 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0)
64	41, 50, 63	divcld 10680	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ)
65		limcresi 23455	. . . . . 6 ⊢ (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶))
66	23	feqmptd 6159	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐹 = (𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)))
67	66	reseq1d 5316	. . . . . . . 8 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (^◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})))
68		difss 3699	. . . . . . . . 9 ⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌
69		resmpt 5369	. . . . . . . . 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	syl6eq 2660	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)))
72	71	oveq1d 6564	. . . . . 6 ⊢ (𝜑 → ((^◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) lim_ℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
73	65, 72	syl5sseq 3616	. . . . 5 ⊢ (𝜑 → (^◡𝐹 lim_ℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
74		f1ocnvfv1 6432	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
75	1, 4, 74	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) = 𝐶)
76		dvcnv.i	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
77	76, 5	cnlimci 23459	. . . . . 6 ⊢ (𝜑 → (^◡𝐹‘(𝐹‘𝐶)) ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
78	75, 77	eqeltrrd 2689	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (^◡𝐹 lim_ℂ (𝐹‘𝐶)))
79	73, 78	sseldd 3569	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (^◡𝐹‘𝑧)) lim_ℂ (𝐹‘𝐶)))
80	45, 35, 4	dvlem 23466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ)
81	37, 40, 52	subne0d 10280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0)
82	50, 41, 63, 81	divne0d 10696	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0)
83		eldifsn 4260	. . . . . . . 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 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}))
85		eqid 2610	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))
86	84, 85	fmptd 6292	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖ {0}))
87		difss 3699	. . . . . . 7 ⊢ (ℂ ∖ {0}) ⊆ ℂ
88	87	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
89		eqid 2610	. . . . . 6 ⊢ (𝐽 ↾_t (ℂ ∖ {0})) = (𝐽 ↾_t (ℂ ∖ {0}))
90	4, 32	eleqtrrd 2691	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
91		dvfg 23476	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
92		ffun 5961	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
93		funfvbrb 6238	. . . . . . . . . 10 ⊢ (Fun (𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
94	9, 91, 92, 93	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
95	90, 94	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))
96	6, 7, 85, 11, 45, 34	eldv 23468	. . . . . . . 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	95, 96	mpbid 221	. . . . . . 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 478	. . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) lim_ℂ 𝐶))
99		resttopon 20775	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾_t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
100	8, 87, 99	mp2an 704	. . . . . . . . 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 9935	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
104	101, 102, 103	cnmptc 21275	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
105	101	cnmptid 21274	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn (𝐽 ↾_t (ℂ ∖ {0}))))
106	7, 89	divcn 22479	. . . . . . . . 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 21279	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ ((𝐽 ↾_t (ℂ ∖ {0})) Cn 𝐽))
109	9, 91	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
110	32	feq2d 5944	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ))
111	109, 110	mpbid 221	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ)
112	111, 4	ffvelrnd 6268	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ)
113		ffn 5958	. . . . . . . . . . 11 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
114	109, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹))
115		fnfvelrn 6264	. . . . . . . . . 10 ⊢ (((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ 𝐶 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
116	114, 90, 115	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹))
117		dvcnv.z	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
118		nelne2 2879	. . . . . . . . 9 ⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
119	116, 117, 118	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0)
120		eldifsn 4260	. . . . . . . 8 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔ (((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0))
121	112, 119, 120	sylanbrc 695	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}))
122	100	toponunii 20547	. . . . . . . 8 ⊢ (ℂ ∖ {0}) = ∪ (𝐽 ↾_t (ℂ ∖ {0}))
123	122	cncnpi 20892	. . . . . . 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)))
124	108, 121, 123	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) ∈ (((𝐽 ↾_t (ℂ ∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶)))
125	86, 88, 7, 89, 98, 124	limccnp 23461	. . . . 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))
126		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶)))
127		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))
128		ovex 6577	. . . . . . 7 ⊢ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ V
129	126, 127, 128	fvmpt 6191	. . . . . 6 ⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶)))
130	121, 129	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶)))
131		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
132		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑥)))
133		oveq2 6557	. . . . . . . 8 ⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))
134	84, 131, 132, 133	fmptco 6303	. . . . . . 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)))))
135	50, 41, 63, 81	recdivd 10697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))
136	135	mpteq2dva 4672	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))))
137	134, 136	eqtrd 2644	. . . . . 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)))))
138	137	oveq1d 6564	. . . . 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))
139	125, 130, 138	3eltr3d 2702	. . . 4 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) lim_ℂ 𝐶))
140		oveq1 6556	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((^◡𝐹‘𝑧) − 𝐶))
141		fveq2 6103	. . . . . 6 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑧)))
142	141	oveq1d 6564	. . . . 5 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))
143	140, 142	oveq12d 6567	. . . 4 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))))
144		eldifsni 4261	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶))
145	144	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶))
146	145	necomd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧)
147	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐹:𝑋–1-1-onto→𝑌)
148	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 ∈ 𝑋)
149	24	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ∈ 𝑌)
150		f1ocnvfvb 6435	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
151	147, 148, 149, 150	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (^◡𝐹‘𝑧) = 𝐶))
152	151	necon3abid 2818	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) ≠ 𝑧 ↔ ¬ (^◡𝐹‘𝑧) = 𝐶))
153	146, 152	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ¬ (^◡𝐹‘𝑧) = 𝐶)
154	153	pm2.21d 117	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) = 𝐶 → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶))))
155	154	impr 647	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (^◡𝐹‘𝑧) = 𝐶)) → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶)))
156	30, 64, 79, 139, 143, 155	limcco 23463	. . 3 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))
157	75	eqcomd 2616	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
158	157	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (^◡𝐹‘(𝐹‘𝐶)))
159	158	oveq2d 6565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((^◡𝐹‘𝑧) − 𝐶) = ((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))))
160		f1ocnvfv2 6433	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
161	1, 24, 160	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
162	161	oveq1d 6564	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶)))
163	159, 162	oveq12d 6567	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (((^◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(^◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶))))
164	163	mpteq2dva 4672	. . . 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)))))
165	164	oveq1d 6564	. . 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)))
166	156, 165	eleqtrd 2690	. 2 ⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((^◡𝐹‘𝑧) − (^◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) lim_ℂ (𝐹‘𝐶)))
167		eqid 2610	. . 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))))
168	23, 35	fssd 5970	. . 3 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
169	6, 7, 167, 11, 168, 43	eldv 23468	. 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)))))
170	20, 166, 169	mpbir2and 959	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 − cmin 10145 / cdiv 10563 ↾_t crest 15904 TopOpenctopn 15905 ℂ_fldccnfld 19567 Topctop 20517 TopOnctopon 20518 intcnt 20631 Cn ccn 20838 CnP ccnp 20839 ×_t ctx 21173 –cn→ccncf 22487 lim_ℂ climc 23432 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: dvcnv 23544

W3C validator