dvacos - Mathbox for Brendan Leahy

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Theorem dvacos 32667

Description: Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.)

**Hypothesis**
Ref	Expression
dvasin.d	⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))

**Assertion**
Ref	Expression
dvacos	⊢ (ℂ D (arccos ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2)))))

Distinct variable group: 𝑥,𝐷

**Proof of Theorem dvacos**
Step	Hyp	Ref	Expression
1		df-acos 24393	. . . . 5 ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
2	1	reseq1i 5313	. . . 4 ⊢ (arccos ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) ↾ 𝐷)
3		dvasin.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
4		difss 3699	. . . . . 6 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⊆ ℂ
5	3, 4	eqsstri 3598	. . . . 5 ⊢ 𝐷 ⊆ ℂ
6		resmpt 5369	. . . . 5 ⊢ (𝐷 ⊆ ℂ → ((𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥))))
7	5, 6	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥)))
8	2, 7	eqtri 2632	. . 3 ⊢ (arccos ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥)))
9	8	oveq2i 6560	. 2 ⊢ (ℂ D (arccos ↾ 𝐷)) = (ℂ D (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥))))
10		cnelprrecn 9908	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
11	10	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
12		halfpire 24020	. . . . . 6 ⊢ (π / 2) ∈ ℝ
13	12	recni 9931	. . . . 5 ⊢ (π / 2) ∈ ℂ
14	13	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (π / 2) ∈ ℂ)
15		c0ex 9913	. . . . 5 ⊢ 0 ∈ V
16	15	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 0 ∈ V)
17	13	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (π / 2) ∈ ℂ)
18	15	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 0 ∈ V)
19	13	a1i 11	. . . . . 6 ⊢ (⊤ → (π / 2) ∈ ℂ)
20	11, 19	dvmptc 23527	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (π / 2))) = (𝑥 ∈ ℂ ↦ 0))
21	5	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ⊆ ℂ)
22		eqid 2610	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	22	cnfldtopon 22396	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
24	23	toponunii 20547	. . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
25	24	restid 15917	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
26	23, 25	ax-mp 5	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
27	26	eqcomi 2619	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
28	22	recld2 22425	. . . . . . . . . 10 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
29		neg1rr 11002	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
30		iocmnfcld 22382	. . . . . . . . . . . 12 ⊢ (-1 ∈ ℝ → (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))))
31	29, 30	ax-mp 5	. . . . . . . . . . 11 ⊢ (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,)))
32	22	tgioo2 22414	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
33	32	fveq2i 6106	. . . . . . . . . . 11 ⊢ (Clsd‘(topGen‘ran (,))) = (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
34	31, 33	eleqtri 2686	. . . . . . . . . 10 ⊢ (-∞(,]-1) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
35		restcldr 20788	. . . . . . . . . 10 ⊢ ((ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)) ∧ (-∞(,]-1) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))) → (-∞(,]-1) ∈ (Clsd‘(TopOpen‘ℂ_fld)))
36	28, 34, 35	mp2an 704	. . . . . . . . 9 ⊢ (-∞(,]-1) ∈ (Clsd‘(TopOpen‘ℂ_fld))
37		1re 9918	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
38		icopnfcld 22381	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ → (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
39	37, 38	ax-mp 5	. . . . . . . . . . 11 ⊢ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))
40	39, 33	eleqtri 2686	. . . . . . . . . 10 ⊢ (1[,)+∞) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
41		restcldr 20788	. . . . . . . . . 10 ⊢ ((ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)) ∧ (1[,)+∞) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))) → (1[,)+∞) ∈ (Clsd‘(TopOpen‘ℂ_fld)))
42	28, 40, 41	mp2an 704	. . . . . . . . 9 ⊢ (1[,)+∞) ∈ (Clsd‘(TopOpen‘ℂ_fld))
43		uncld 20655	. . . . . . . . 9 ⊢ (((-∞(,]-1) ∈ (Clsd‘(TopOpen‘ℂ_fld)) ∧ (1[,)+∞) ∈ (Clsd‘(TopOpen‘ℂ_fld))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld)))
44	36, 42, 43	mp2an 704	. . . . . . . 8 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld))
45	24	cldopn 20645	. . . . . . . 8 ⊢ (((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld)) → (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂ_fld))
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂ_fld)
47	3, 46	eqeltri 2684	. . . . . 6 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
48	47	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
49	11, 17, 18, 20, 21, 27, 22, 48	dvmptres 23532	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (π / 2))) = (𝑥 ∈ 𝐷 ↦ 0))
50	5	sseli 3564	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
51		asincl 24400	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (arcsin‘𝑥) ∈ ℂ)
52	50, 51	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (arcsin‘𝑥) ∈ ℂ)
53	52	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (arcsin‘𝑥) ∈ ℂ)
54		ovex 6577	. . . . 5 ⊢ (1 / (√‘(1 − (𝑥↑2)))) ∈ V
55	54	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (1 / (√‘(1 − (𝑥↑2)))) ∈ V)
56	3	dvasin 32666	. . . . 5 ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))
57		asinf 24399	. . . . . . . 8 ⊢ arcsin:ℂ⟶ℂ
58	57	a1i 11	. . . . . . 7 ⊢ (⊤ → arcsin:ℂ⟶ℂ)
59	58, 21	feqresmpt 6160	. . . . . 6 ⊢ (⊤ → (arcsin ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (arcsin‘𝑥)))
60	59	oveq2d 6565	. . . . 5 ⊢ (⊤ → (ℂ D (arcsin ↾ 𝐷)) = (ℂ D (𝑥 ∈ 𝐷 ↦ (arcsin‘𝑥))))
61	56, 60	syl5reqr 2659	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (arcsin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2))))))
62	11, 14, 16, 49, 53, 55, 61	dvmptsub 23536	. . 3 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (0 − (1 / (√‘(1 − (𝑥↑2)))))))
63	62	trud 1484	. 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ ((π / 2) − (arcsin‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (0 − (1 / (√‘(1 − (𝑥↑2))))))
64		df-neg 10148	. . . 4 ⊢ -(1 / (√‘(1 − (𝑥↑2)))) = (0 − (1 / (√‘(1 − (𝑥↑2)))))
65		1cnd 9935	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 1 ∈ ℂ)
66		ax-1cn 9873	. . . . . . 7 ⊢ 1 ∈ ℂ
67	50	sqcld 12868	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑2) ∈ ℂ)
68		subcl 10159	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 − (𝑥↑2)) ∈ ℂ)
69	66, 67, 68	sylancr 694	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (1 − (𝑥↑2)) ∈ ℂ)
70	69	sqrtcld 14024	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (√‘(1 − (𝑥↑2))) ∈ ℂ)
71		eldifn 3695	. . . . . . . 8 ⊢ (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
72	71, 3	eleq2s 2706	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
73		mnfxr 9975	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
74	29	rexri 9976	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ^*
75		mnflt 11833	. . . . . . . . . . . . 13 ⊢ (-1 ∈ ℝ → -∞ < -1)
76	29, 75	ax-mp 5	. . . . . . . . . . . 12 ⊢ -∞ < -1
77		ubioc1 12098	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ -∞ < -1) → -1 ∈ (-∞(,]-1))
78	73, 74, 76, 77	mp3an 1416	. . . . . . . . . . 11 ⊢ -1 ∈ (-∞(,]-1)
79		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑥 = -1 → (𝑥 ∈ (-∞(,]-1) ↔ -1 ∈ (-∞(,]-1)))
80	78, 79	mpbiri 247	. . . . . . . . . 10 ⊢ (𝑥 = -1 → 𝑥 ∈ (-∞(,]-1))
81	37	rexri 9976	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ^*
82		pnfxr 9971	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
83		ltpnf 11830	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ → 1 < +∞)
84	37, 83	ax-mp 5	. . . . . . . . . . . 12 ⊢ 1 < +∞
85		lbico1 12099	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 1 < +∞) → 1 ∈ (1[,)+∞))
86	81, 82, 84, 85	mp3an 1416	. . . . . . . . . . 11 ⊢ 1 ∈ (1[,)+∞)
87		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 ∈ (1[,)+∞) ↔ 1 ∈ (1[,)+∞)))
88	86, 87	mpbiri 247	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 ∈ (1[,)+∞))
89	80, 88	orim12i 537	. . . . . . . . 9 ⊢ ((𝑥 = -1 ∨ 𝑥 = 1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
90	89	orcoms 403	. . . . . . . 8 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
91		elun 3715	. . . . . . . 8 ⊢ (𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)) ↔ (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
92	90, 91	sylibr 223	. . . . . . 7 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
93	72, 92	nsyl 134	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ¬ (𝑥 = 1 ∨ 𝑥 = -1))
94		sq1 12820	. . . . . . . . . 10 ⊢ (1↑2) = 1
95		1cnd 9935	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 ∈ ℂ)
96		sqcl 12787	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
97	96	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) ∈ ℂ)
98	66, 96, 68	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (1 − (𝑥↑2)) ∈ ℂ)
99	98	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) ∈ ℂ)
100		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (√‘(1 − (𝑥↑2))) = 0)
101	99, 100	sqr00d 14028	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) = 0)
102	95, 97, 101	subeq0d 10279	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 = (𝑥↑2))
103	94, 102	syl5req 2657	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) = (1↑2))
104	103	ex 449	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥↑2) = (1↑2)))
105		sqeqor 12840	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
106	66, 105	mpan2 703	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
107	104, 106	sylibd 228	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥 = 1 ∨ 𝑥 = -1)))
108	107	necon3bd 2796	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (¬ (𝑥 = 1 ∨ 𝑥 = -1) → (√‘(1 − (𝑥↑2))) ≠ 0))
109	50, 93, 108	sylc 63	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (√‘(1 − (𝑥↑2))) ≠ 0)
110	65, 70, 109	divnegd 10693	. . . 4 ⊢ (𝑥 ∈ 𝐷 → -(1 / (√‘(1 − (𝑥↑2)))) = (-1 / (√‘(1 − (𝑥↑2)))))
111	64, 110	syl5eqr 2658	. . 3 ⊢ (𝑥 ∈ 𝐷 → (0 − (1 / (√‘(1 − (𝑥↑2))))) = (-1 / (√‘(1 − (𝑥↑2)))))
112	111	mpteq2ia 4668	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (0 − (1 / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2)))))
113	9, 63, 112	3eqtri 2636	1 ⊢ (ℂ D (arccos ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 (,)cioo 12046 (,]cioc 12047 [,)cico 12048 ↑cexp 12722 √csqrt 13821 πcpi 14636 ↾_t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 TopOnctopon 20518 Clsdccld 20630 D cdv 23433 arcsincasin 24389 arccoscacos 24390

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-addf 9894 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-fal 1481 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-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 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-cmp 21000 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 df-log 24107 df-cxp 24108 df-asin 24392 df-acos 24393

This theorem is referenced by: dvreacos 32669

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