dvasin - Mathbox for Brendan Leahy

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Theorem dvasin 35141

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

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

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

Distinct variable group: 𝑥,𝐷

Proof of Theorem dvasin Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-asin 25451	. . . . 5 ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
2	1	reseq1i 5814	. . . 4 ⊢ (arcsin ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) ↾ 𝐷)
3		dvasin.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
4		difss 4059	. . . . . 6 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⊆ ℂ
5	3, 4	eqsstri 3949	. . . . 5 ⊢ 𝐷 ⊆ ℂ
6		resmpt 5872	. . . . 5 ⊢ (𝐷 ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))))
7	5, 6	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
8	2, 7	eqtri 2821	. . 3 ⊢ (arcsin ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
9	8	oveq2i 7146	. 2 ⊢ (ℂ D (arcsin ↾ 𝐷)) = (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))))
10		cnelprrecn 10619	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
11	10	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
12	5	sseli 3911	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
13		ax-icn 10585	. . . . . . . . 9 ⊢ i ∈ ℂ
14		mulcl 10610	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ)
15	13, 14	mpan 689	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (i · 𝑥) ∈ ℂ)
16		ax-1cn 10584	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
17		sqcl 13480	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
18		subcl 10874	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 − (𝑥↑2)) ∈ ℂ)
19	16, 17, 18	sylancr 590	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 − (𝑥↑2)) ∈ ℂ)
20	19	sqrtcld 14789	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (√‘(1 − (𝑥↑2))) ∈ ℂ)
21	15, 20	addcld 10649	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℂ)
22	12, 21	syl 17	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℂ)
23		asinlem 25454	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≠ 0)
24	12, 23	syl 17	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≠ 0)
25	22, 24	logcld 25162	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
26	25	adantl 485	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
27		ovexd 7170	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (i / (√‘(1 − (𝑥↑2)))) ∈ V)
28		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)
29		asinlem3 25457	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → 0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
30		rere 14473	. . . . . . . . . . . . . . . . . . 19 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
31	30	breq2d 5042	. . . . . . . . . . . . . . . . . 18 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ↔ 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
32	31	biimpac 482	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
33	29, 32	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
34	23	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≠ 0)
35	28, 33, 34	ne0gt0d 10766	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
36		0re 10632	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
37		ltnle 10709	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → (0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ↔ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
38	36, 37	mpan 689	. . . . . . . . . . . . . . . 16 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ↔ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
39	38	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → (0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ↔ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
40	35, 39	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0)
41	40	ex 416	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
42	12, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
43		imor 850	. . . . . . . . . . . 12 ⊢ ((((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0) ↔ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
44	42, 43	sylib 221	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
45	44	orcomd 868	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ))
46	45	olcd 871	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)))
47		3ianor 1104	. . . . . . . . . . 11 ⊢ (¬ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∧ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0) ↔ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∨ ¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
48		3orrot 1089	. . . . . . . . . . 11 ⊢ ((¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∨ ¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0) ↔ (¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ))
49		3orass 1087	. . . . . . . . . . 11 ⊢ ((¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) ↔ (¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)))
50	47, 48, 49	3bitrri 301	. . . . . . . . . 10 ⊢ ((¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)) ↔ ¬ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∧ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
51		mnfxr 10687	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
52		elioc2 12788	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (-∞(,]0) ↔ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∧ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0)))
53	51, 36, 52	mp2an 691	. . . . . . . . . 10 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (-∞(,]0) ↔ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∧ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
54	50, 53	xchbinxr 338	. . . . . . . . 9 ⊢ ((¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)) ↔ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (-∞(,]0))
55	46, 54	sylib 221	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (-∞(,]0))
56	22, 55	eldifd 3892	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (ℂ ∖ (-∞(,]0)))
57	56	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (ℂ ∖ (-∞(,]0)))
58		ovexd 7170	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))) ∈ V)
59		eldifi 4054	. . . . . . . 8 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
60		eldifn 4055	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
61		0xr 10677	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
62		mnflt0 12508	. . . . . . . . . . . 12 ⊢ -∞ < 0
63		ubioc1 12778	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
64	51, 61, 62, 63	mp3an 1458	. . . . . . . . . . 11 ⊢ 0 ∈ (-∞(,]0)
65		eleq1 2877	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
66	64, 65	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
67	66	necon3bi 3013	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (-∞(,]0) → 𝑦 ≠ 0)
68	60, 67	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ≠ 0)
69	59, 68	logcld 25162	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → (log‘𝑦) ∈ ℂ)
70	69	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
71		ovexd 7170	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
72	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → i ∈ ℂ)
73	72, 12	mulcld 10650	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (i · 𝑥) ∈ ℂ)
74	73	adantl 485	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (i · 𝑥) ∈ ℂ)
75	13	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → i ∈ ℂ)
76	12	adantl 485	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
77		1cnd 10625	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 1 ∈ ℂ)
78		simpr 488	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
79		1cnd 10625	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
80	11	dvmptid 24560	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
81	5	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 𝐷 ⊆ ℂ)
82		eqid 2798	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
83	82	cnfldtopon 23388	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
84	83	toponrestid 21526	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
85	82	recld2 23419	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
86		neg1rr 11740	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℝ
87		iocmnfcld 23374	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ∈ ℝ → (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))))
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,)))
89		1re 10630	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
90		icopnfcld 23373	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
91	89, 90	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))
92		uncld 21646	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))) ∧ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,))))
93	88, 91, 92	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,)))
94	82	tgioo2 23408	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
95	94	fveq2i 6648	. . . . . . . . . . . . . . . 16 ⊢ (Clsd‘(topGen‘ran (,))) = (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
96	93, 95	eleqtri 2888	. . . . . . . . . . . . . . 15 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
97		restcldr 21779	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld)) ∧ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld)))
98	85, 96, 97	mp2an 691	. . . . . . . . . . . . . 14 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld))
99	83	toponunii 21521	. . . . . . . . . . . . . . 15 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
100	99	cldopn 21636	. . . . . . . . . . . . . 14 ⊢ (((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld)) → (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂ_fld))
101	98, 100	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂ_fld)
102	3, 101	eqeltri 2886	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
103	102	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
104	11, 78, 79, 80, 81, 84, 82, 103	dvmptres 24566	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ 1))
105	13	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → i ∈ ℂ)
106	11, 76, 77, 104, 105	dvmptcmul 24567	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (i · 𝑥))) = (𝑥 ∈ 𝐷 ↦ (i · 1)))
107	13	mulid1i 10634	. . . . . . . . . 10 ⊢ (i · 1) = i
108	107	mpteq2i 5122	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↦ (i · 1)) = (𝑥 ∈ 𝐷 ↦ i)
109	106, 108	eqtrdi 2849	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (i · 𝑥))) = (𝑥 ∈ 𝐷 ↦ i))
110	12	sqcld 13504	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑2) ∈ ℂ)
111	16, 110, 18	sylancr 590	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (1 − (𝑥↑2)) ∈ ℂ)
112	111	sqrtcld 14789	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (√‘(1 − (𝑥↑2))) ∈ ℂ)
113	112	adantl 485	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘(1 − (𝑥↑2))) ∈ ℂ)
114		ovexd 7170	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (-𝑥 / (√‘(1 − (𝑥↑2)))) ∈ V)
115		elin 3897	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐷 ∩ ℝ) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ))
116	3	asindmre 35140	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)
117	116	eqimssi 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∩ ℝ) ⊆ (-1(,)1)
118	117	sseli 3911	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐷 ∩ ℝ) → 𝑥 ∈ (-1(,)1))
119	115, 118	sylbir 238	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ (-1(,)1))
120		incom 4128	. . . . . . . . . . . . . . . 16 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ((-∞(,]0) ∩ (0(,)+∞))
121		pnfxr 10684	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
122		df-ioc 12731	. . . . . . . . . . . . . . . . . 18 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
123		df-ioo 12730	. . . . . . . . . . . . . . . . . 18 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
124		xrltnle 10697	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
125	122, 123, 124	ixxdisj 12741	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
126	51, 61, 121, 125	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
127	120, 126	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
128		elioore 12756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (-1(,)1) → 𝑥 ∈ ℝ)
129	128	resqcld 13607	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (-1(,)1) → (𝑥↑2) ∈ ℝ)
130		resubcl 10939	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (1 − (𝑥↑2)) ∈ ℝ)
131	89, 129, 130	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ ℝ)
132	86	rexri 10688	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ^*
133		1xr 10689	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
134		elioo2 12767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝑥 ∈ (-1(,)1) ↔ (𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1)))
135	132, 133, 134	mp2an 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (-1(,)1) ↔ (𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1))
136		recn 10616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
137	136	abscld 14788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
138	136	absge0d 14796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → 0 ≤ (abs‘𝑥))
139		0le1 11152	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ≤ 1
140		lt2sq 13494	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (1 ∈ ℝ ∧ 0 ≤ 1)) → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
141	89, 139, 140	mpanr12 704	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
142	137, 138, 141	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
143		abslt 14666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑥) < 1 ↔ (-1 < 𝑥 ∧ 𝑥 < 1)))
144	89, 143	mpan2 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥) < 1 ↔ (-1 < 𝑥 ∧ 𝑥 < 1)))
145		absresq 14654	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥)↑2) = (𝑥↑2))
146		sq1 13554	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1↑2) = 1
147	146	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → (1↑2) = 1)
148	145, 147	breq12d 5043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (((abs‘𝑥)↑2) < (1↑2) ↔ (𝑥↑2) < 1))
149		resqcl 13486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → (𝑥↑2) ∈ ℝ)
150		posdif 11122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥↑2) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑥↑2) < 1 ↔ 0 < (1 − (𝑥↑2))))
151	149, 89, 150	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → ((𝑥↑2) < 1 ↔ 0 < (1 − (𝑥↑2))))
152	148, 151	bitrd 282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → (((abs‘𝑥)↑2) < (1↑2) ↔ 0 < (1 − (𝑥↑2))))
153	142, 144, 152	3bitr3d 312	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ → ((-1 < 𝑥 ∧ 𝑥 < 1) ↔ 0 < (1 − (𝑥↑2))))
154	153	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → ((-1 < 𝑥 ∧ 𝑥 < 1) → 0 < (1 − (𝑥↑2))))
155	154	3impib 1113	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1) → 0 < (1 − (𝑥↑2)))
156	135, 155	sylbi 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-1(,)1) → 0 < (1 − (𝑥↑2)))
157	131, 156	elrpd 12416	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ ℝ⁺)
158		ioorp 12803	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
159	157, 158	eleqtrrdi 2901	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ (0(,)+∞))
160		disjel 4364	. . . . . . . . . . . . . . 15 ⊢ ((((0(,)+∞) ∩ (-∞(,]0)) = ∅ ∧ (1 − (𝑥↑2)) ∈ (0(,)+∞)) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
161	127, 159, 160	sylancr 590	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (-1(,)1) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
162	119, 161	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
163		elioc2 12788	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → ((1 − (𝑥↑2)) ∈ (-∞(,]0) ↔ ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0)))
164	51, 36, 163	mp2an 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) ↔ ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0))
165	164	biimpi 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0))
166	165	simp1d 1139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (𝑥↑2)) ∈ ℝ)
167		resubcl 10939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ ∧ (1 − (𝑥↑2)) ∈ ℝ) → (1 − (1 − (𝑥↑2))) ∈ ℝ)
168	89, 166, 167	sylancr 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (1 − (𝑥↑2))) ∈ ℝ)
169		nncan 10904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
170	16, 169	mpan 689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥↑2) ∈ ℂ → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
171	170	eleq1d 2874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥↑2) ∈ ℂ → ((1 − (1 − (𝑥↑2))) ∈ ℝ ↔ (𝑥↑2) ∈ ℝ))
172	171	biimpa 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (1 − (𝑥↑2))) ∈ ℝ) → (𝑥↑2) ∈ ℝ)
173	168, 172	sylan2 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥↑2) ∈ ℝ)
174	166	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (𝑥↑2)) ∈ ℝ)
175	165	simp3d 1141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (𝑥↑2)) ≤ 0)
176	175	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (𝑥↑2)) ≤ 0)
177		letr 10723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 − (𝑥↑2)) ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ) → (((1 − (𝑥↑2)) ≤ 0 ∧ 0 ≤ 1) → (1 − (𝑥↑2)) ≤ 1))
178	36, 89, 177	mp3an23 1450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ ℝ → (((1 − (𝑥↑2)) ≤ 0 ∧ 0 ≤ 1) → (1 − (𝑥↑2)) ≤ 1))
179	139, 178	mpan2i 696	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 − (𝑥↑2)) ∈ ℝ → ((1 − (𝑥↑2)) ≤ 0 → (1 − (𝑥↑2)) ≤ 1))
180	174, 176, 179	sylc 65	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (𝑥↑2)) ≤ 1)
181		subge0 11142	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℝ ∧ (1 − (𝑥↑2)) ∈ ℝ) → (0 ≤ (1 − (1 − (𝑥↑2))) ↔ (1 − (𝑥↑2)) ≤ 1))
182	89, 174, 181	sylancr 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (0 ≤ (1 − (1 − (𝑥↑2))) ↔ (1 − (𝑥↑2)) ≤ 1))
183	180, 182	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 0 ≤ (1 − (1 − (𝑥↑2))))
184	170	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
185	183, 184	breqtrd 5056	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 0 ≤ (𝑥↑2))
186	173, 185	resqrtcld 14769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (√‘(𝑥↑2)) ∈ ℝ)
187	17, 186	sylan 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (√‘(𝑥↑2)) ∈ ℝ)
188		eleq1 2877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (√‘(𝑥↑2)) → (𝑥 ∈ ℝ ↔ (√‘(𝑥↑2)) ∈ ℝ))
189	187, 188	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = (√‘(𝑥↑2)) → 𝑥 ∈ ℝ))
190	187	renegcld 11056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → -(√‘(𝑥↑2)) ∈ ℝ)
191		eleq1 2877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -(√‘(𝑥↑2)) → (𝑥 ∈ ℝ ↔ -(√‘(𝑥↑2)) ∈ ℝ))
192	190, 191	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = -(√‘(𝑥↑2)) → 𝑥 ∈ ℝ))
193		eqid 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥↑2) = (𝑥↑2)
194		eqsqrtor 14718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → ((𝑥↑2) = (𝑥↑2) ↔ (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2)))))
195	17, 194	mpdan 686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = (𝑥↑2) ↔ (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2)))))
196	193, 195	mpbii 236	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2))))
197	196	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2))))
198	189, 192, 197	mpjaod 857	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 𝑥 ∈ ℝ)
199	198	stoic1a 1774	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ ¬ 𝑥 ∈ ℝ) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
200	12, 199	sylan 583	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ ℝ) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
201	162, 200	pm2.61dan 812	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
202	111, 201	eldifd 3892	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (1 − (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0)))
203	202	adantl 485	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (1 − (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0)))
204		2cnd 11703	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 2 ∈ ℂ)
205		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
206	204, 205	mulcld 10650	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (2 · 𝑥) ∈ ℂ)
207	206	negcld 10973	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → -(2 · 𝑥) ∈ ℂ)
208	207	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(2 · 𝑥) ∈ ℂ)
209	12, 208	sylan2 595	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → -(2 · 𝑥) ∈ ℂ)
210	59	sqrtcld 14789	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → (√‘𝑦) ∈ ℂ)
211	210	adantl 485	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (√‘𝑦) ∈ ℂ)
212		ovexd 7170	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / (2 · (√‘𝑦))) ∈ V)
213	19	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1 − (𝑥↑2)) ∈ ℂ)
214	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 0 ∈ ℝ)
215		1cnd 10625	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℂ)
216	11, 215	dvmptc 24561	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ 1)) = (𝑥 ∈ ℂ ↦ 0))
217	17	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
218		2cn 11700	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
219		mulcl 10610	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
220	218, 219	mpan 689	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (2 · 𝑥) ∈ ℂ)
221	220	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
222		2nn 11698	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
223		dvexp 24556	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))))
224	222, 223	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1))))
225		2m1e1 11751	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 − 1) = 1
226	225	oveq2i 7146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
227		exp1 13431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
228	226, 227	syl5eq 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑥↑(2 − 1)) = 𝑥)
229	228	oveq2d 7151	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
230	229	mpteq2ia 5121	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥))
231	224, 230	eqtri 2821	. . . . . . . . . . . . . 14 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥))
232	231	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑2))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥)))
233	11, 79, 214, 216, 217, 221, 232	dvmptsub 24570	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ ℂ ↦ (0 − (2 · 𝑥))))
234		df-neg 10862	. . . . . . . . . . . . 13 ⊢ -(2 · 𝑥) = (0 − (2 · 𝑥))
235	234	mpteq2i 5122	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(2 · 𝑥)) = (𝑥 ∈ ℂ ↦ (0 − (2 · 𝑥)))
236	233, 235	eqtr4di 2851	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ ℂ ↦ -(2 · 𝑥)))
237	11, 213, 208, 236, 81, 84, 82, 103	dvmptres 24566	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ 𝐷 ↦ -(2 · 𝑥)))
238		eqid 2798	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
239	238	dvcnsqrt 25333	. . . . . . . . . . 11 ⊢ (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (√‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / (2 · (√‘𝑦))))
240	239	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (√‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / (2 · (√‘𝑦)))))
241		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑦 = (1 − (𝑥↑2)) → (√‘𝑦) = (√‘(1 − (𝑥↑2))))
242	241	oveq2d 7151	. . . . . . . . . . 11 ⊢ (𝑦 = (1 − (𝑥↑2)) → (2 · (√‘𝑦)) = (2 · (√‘(1 − (𝑥↑2)))))
243	242	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝑦 = (1 − (𝑥↑2)) → (1 / (2 · (√‘𝑦))) = (1 / (2 · (√‘(1 − (𝑥↑2))))))
244	11, 11, 203, 209, 211, 212, 237, 240, 241, 243	dvmptco 24575	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘(1 − (𝑥↑2))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥))))
245		mulneg2 11066	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · -𝑥) = -(2 · 𝑥))
246	218, 12, 245	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · -𝑥) = -(2 · 𝑥))
247	246	oveq1d 7150	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-(2 · 𝑥) / (2 · (√‘(1 − (𝑥↑2))))))
248	12	negcld 10973	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → -𝑥 ∈ ℂ)
249		eldifn 4055	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
250	249, 3	eleq2s 2908	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
251		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = -1 → 𝑥 = -1)
252		mnflt 12506	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-1 ∈ ℝ → -∞ < -1)
253	86, 252	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ < -1
254		ubioc1 12778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ -∞ < -1) → -1 ∈ (-∞(,]-1))
255	51, 132, 253, 254	mp3an 1458	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ (-∞(,]-1)
256	251, 255	eqeltrdi 2898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -1 → 𝑥 ∈ (-∞(,]-1))
257		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 1 → 𝑥 = 1)
258		ltpnf 12503	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
259	89, 258	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
260		lbico1 12779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 1 < +∞) → 1 ∈ (1[,)+∞))
261	133, 121, 259, 260	mp3an 1458	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ (1[,)+∞)
262	257, 261	eqeltrdi 2898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → 𝑥 ∈ (1[,)+∞))
263	256, 262	orim12i 906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = -1 ∨ 𝑥 = 1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
264	263	orcoms 869	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
265		elun 4076	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)) ↔ (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
266	264, 265	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
267	250, 266	nsyl 142	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → ¬ (𝑥 = 1 ∨ 𝑥 = -1))
268		1cnd 10625	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 ∈ ℂ)
269	17	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) ∈ ℂ)
270	19	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) ∈ ℂ)
271		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (√‘(1 − (𝑥↑2))) = 0)
272	270, 271	sqr00d 14793	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) = 0)
273	268, 269, 272	subeq0d 10994	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 = (𝑥↑2))
274	146, 273	syl5req 2846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) = (1↑2))
275	274	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥↑2) = (1↑2)))
276		sqeqor 13574	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
277	16, 276	mpan2 690	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
278	275, 277	sylibd 242	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥 = 1 ∨ 𝑥 = -1)))
279	278	necon3bd 3001	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (¬ (𝑥 = 1 ∨ 𝑥 = -1) → (√‘(1 − (𝑥↑2))) ≠ 0))
280	12, 267, 279	sylc 65	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (√‘(1 − (𝑥↑2))) ≠ 0)
281		2cnne0 11835	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
282		divcan5 11331	. . . . . . . . . . . . 13 ⊢ ((-𝑥 ∈ ℂ ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
283	281, 282	mp3an3 1447	. . . . . . . . . . . 12 ⊢ ((-𝑥 ∈ ℂ ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0)) → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
284	248, 112, 280, 283	syl12anc 835	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
285	218, 12, 219	sylancr 590	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → (2 · 𝑥) ∈ ℂ)
286	285	negcld 10973	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → -(2 · 𝑥) ∈ ℂ)
287		mulcl 10610	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ∈ ℂ) → (2 · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
288	218, 112, 287	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
289		mulne0 11271	. . . . . . . . . . . . . 14 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0)) → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
290	281, 289	mpan 689	. . . . . . . . . . . . 13 ⊢ (((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0) → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
291	112, 280, 290	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
292	286, 288, 291	divrec2d 11409	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-(2 · 𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥)))
293	247, 284, 292	3eqtr3rd 2842	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥)) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
294	293	mpteq2ia 5121	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥))) = (𝑥 ∈ 𝐷 ↦ (-𝑥 / (√‘(1 − (𝑥↑2)))))
295	244, 294	eqtrdi 2849	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘(1 − (𝑥↑2))))) = (𝑥 ∈ 𝐷 ↦ (-𝑥 / (√‘(1 − (𝑥↑2))))))
296	11, 74, 75, 109, 113, 114, 295	dvmptadd 24563	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (i + (-𝑥 / (√‘(1 − (𝑥↑2)))))))
297		mulcl 10610	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ∈ ℂ) → (i · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
298	13, 20, 297	sylancr 590	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (i · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
299	12, 298	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (i · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
300	299, 248, 112, 280	divdird 11443	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((i · (√‘(1 − (𝑥↑2)))) + -𝑥) / (√‘(1 − (𝑥↑2)))) = (((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) + (-𝑥 / (√‘(1 − (𝑥↑2))))))
301		ixi 11258	. . . . . . . . . . . . . . . 16 ⊢ (i · i) = -1
302	301	eqcomi 2807	. . . . . . . . . . . . . . 15 ⊢ -1 = (i · i)
303	302	oveq1i 7145	. . . . . . . . . . . . . 14 ⊢ (-1 · 𝑥) = ((i · i) · 𝑥)
304		mulm1 11070	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (-1 · 𝑥) = -𝑥)
305		mulass 10614	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((i · i) · 𝑥) = (i · (i · 𝑥)))
306	13, 13, 305	mp3an12 1448	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((i · i) · 𝑥) = (i · (i · 𝑥)))
307	303, 304, 306	3eqtr3a 2857	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → -𝑥 = (i · (i · 𝑥)))
308	307	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (-𝑥 + (i · (√‘(1 − (𝑥↑2))))) = ((i · (i · 𝑥)) + (i · (√‘(1 − (𝑥↑2))))))
309		negcl 10875	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ)
310	298, 309	addcomd 10831	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((i · (√‘(1 − (𝑥↑2)))) + -𝑥) = (-𝑥 + (i · (√‘(1 − (𝑥↑2))))))
311	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → i ∈ ℂ)
312	311, 15, 20	adddid 10654	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = ((i · (i · 𝑥)) + (i · (√‘(1 − (𝑥↑2))))))
313	308, 310, 312	3eqtr4d 2843	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((i · (√‘(1 − (𝑥↑2)))) + -𝑥) = (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
314	12, 313	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ((i · (√‘(1 − (𝑥↑2)))) + -𝑥) = (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
315	314	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((i · (√‘(1 − (𝑥↑2)))) + -𝑥) / (√‘(1 − (𝑥↑2)))) = ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))
316	72, 112, 280	divcan4d 11411	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) = i)
317	316	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) + (-𝑥 / (√‘(1 − (𝑥↑2))))) = (i + (-𝑥 / (√‘(1 − (𝑥↑2))))))
318	300, 315, 317	3eqtr3rd 2842	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (i + (-𝑥 / (√‘(1 − (𝑥↑2))))) = ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))
319	318	mpteq2ia 5121	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ (i + (-𝑥 / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))
320	296, 319	eqtrdi 2849	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2))))))
321	238	dvlog 25242	. . . . . . 7 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
322		logf1o 25156	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
323		f1of 6590	. . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
324	322, 323	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → log:(ℂ ∖ {0})⟶ran log)
325		snssi 4701	. . . . . . . . . . 11 ⊢ (0 ∈ (-∞(,]0) → {0} ⊆ (-∞(,]0))
326	64, 325	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ (-∞(,]0)
327		sscon 4066	. . . . . . . . . 10 ⊢ ({0} ⊆ (-∞(,]0) → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
328	326, 327	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
329	324, 328	feqresmpt 6709	. . . . . . . 8 ⊢ (⊤ → (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
330	329	oveq2d 7151	. . . . . . 7 ⊢ (⊤ → (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))))
331	321, 330	syl5reqr 2848	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦)))
332		fveq2 6645	. . . . . 6 ⊢ (𝑦 = ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) → (log‘𝑦) = (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
333		oveq2 7143	. . . . . 6 ⊢ (𝑦 = ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) → (1 / 𝑦) = (1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
334	11, 11, 57, 58, 70, 71, 320, 331, 332, 333	dvmptco 24575	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) = (𝑥 ∈ 𝐷 ↦ ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))))
335	22, 24	reccld 11398	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
336		mulcl 10610	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℂ) → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
337	13, 21, 336	sylancr 590	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
338	12, 337	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
339	335, 338, 112, 280	divassd 11440	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) / (√‘(1 − (𝑥↑2)))) = ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2))))))
340	338, 22, 24	divrec2d 11409	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
341	72, 22, 24	divcan4d 11411	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = i)
342	340, 341	eqtr3d 2835	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = i)
343	342	oveq1d 7150	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) / (√‘(1 − (𝑥↑2)))) = (i / (√‘(1 − (𝑥↑2)))))
344	339, 343	eqtr3d 2835	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2))))) = (i / (√‘(1 − (𝑥↑2)))))
345	344	mpteq2ia 5121	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (i / (√‘(1 − (𝑥↑2)))))
346	334, 345	eqtrdi 2849	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) = (𝑥 ∈ 𝐷 ↦ (i / (√‘(1 − (𝑥↑2))))))
347		negicn 10876	. . . . 5 ⊢ -i ∈ ℂ
348	347	a1i 11	. . . 4 ⊢ (⊤ → -i ∈ ℂ)
349	11, 26, 27, 346, 348	dvmptcmul 24567	. . 3 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))) = (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2)))))))
350	349	mptru 1545	. 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))) = (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2))))))
351	13, 13	mulneg1i 11075	. . . . . 6 ⊢ (-i · i) = -(i · i)
352	301	negeqi 10868	. . . . . 6 ⊢ -(i · i) = --1
353		negneg1e1 11743	. . . . . 6 ⊢ --1 = 1
354	351, 352, 353	3eqtri 2825	. . . . 5 ⊢ (-i · i) = 1
355	354	oveq1i 7145	. . . 4 ⊢ ((-i · i) / (√‘(1 − (𝑥↑2)))) = (1 / (√‘(1 − (𝑥↑2))))
356		divass 11305	. . . . . 6 ⊢ ((-i ∈ ℂ ∧ i ∈ ℂ ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0)) → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
357	347, 13, 356	mp3an12 1448	. . . . 5 ⊢ (((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0) → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
358	112, 280, 357	syl2anc 587	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
359	355, 358	syl5reqr 2848	. . 3 ⊢ (𝑥 ∈ 𝐷 → (-i · (i / (√‘(1 − (𝑥↑2))))) = (1 / (√‘(1 − (𝑥↑2)))))
360	359	mpteq2ia 5121	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))
361	9, 350, 360	3eqtri 2825	1 ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 {cpr 4527 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℝ⁺crp 12377 (,)cioo 12726 (,]cioc 12727 [,)cico 12728 ↑cexp 13425 ℜcre 14448 √csqrt 14584 abscabs 14585 ↾_t crest 16686 TopOpenctopn 16687 topGenctg 16703 ℂ_fldccnfld 20091 Clsdccld 21621 D cdv 24466 logclog 25146 arcsincasin 25448

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 df-asin 25451

This theorem is referenced by: dvacos 35142 dvreasin 35143

W3C validator