dvasin - Mathbox for Brendan Leahy

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Theorem dvasin 37765

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 26803	. . . . 5 ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
2	1	reseq1i 5928	. . . 4 ⊢ (arcsin ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) ↾ 𝐷)
3		dvasin.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
4		difss 4085	. . . . . 6 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⊆ ℂ
5	3, 4	eqsstri 3977	. . . . 5 ⊢ 𝐷 ⊆ ℂ
6		resmpt 5990	. . . . 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 2756	. . 3 ⊢ (arcsin ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
9	8	oveq2i 7363	. 2 ⊢ (ℂ D (arcsin ↾ 𝐷)) = (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))))
10		cnelprrecn 11106	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
11	10	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
12	5	sseli 3926	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
13		ax-icn 11072	. . . . . . . . 9 ⊢ i ∈ ℂ
14		mulcl 11097	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ)
15	13, 14	mpan 690	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (i · 𝑥) ∈ ℂ)
16		ax-1cn 11071	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
17		sqcl 14027	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
18		subcl 11366	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 − (𝑥↑2)) ∈ ℂ)
19	16, 17, 18	sylancr 587	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 − (𝑥↑2)) ∈ ℂ)
20	19	sqrtcld 15349	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (√‘(1 − (𝑥↑2))) ∈ ℂ)
21	15, 20	addcld 11138	. . . . . . 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 26806	. . . . . . 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 26507	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
26	25	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
27		ovexd 7387	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (i / (√‘(1 − (𝑥↑2)))) ∈ V)
28		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)
29		asinlem3 26809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → 0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
30		rere 15031	. . . . . . . . . . . . . . . . . . 19 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
31	30	breq2d 5105	. . . . . . . . . . . . . . . . . 18 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ↔ 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
32	31	biimpac 478	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ≤ (ℜ‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
33	29, 32	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 ≤ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
34	23	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≠ 0)
35	28, 33, 34	ne0gt0d 11257	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → 0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))
36		0re 11121	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
37		ltnle 11199	. . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . 16 ⊢ (((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ → (0 < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ↔ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
39	38	adantl 481	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ) → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0)
41	40	ex 412	. . . . . . . . . . . . 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 853	. . . . . . . . . . . 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 218	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0))
45	44	orcomd 871	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ))
46	45	olcd 874	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (¬ -∞ < ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∨ (¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ≤ 0 ∨ ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℝ)))
47		3ianor 1106	. . . . . . . . . . 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 1091	. . . . . . . . . . 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 1089	. . . . . . . . . . 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 298	. . . . . . . . . 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 11176	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
52		elioc2 13311	. . . . . . . . . . 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 692	. . . . . . . . . 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 335	. . . . . . . . 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 218	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ¬ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (-∞(,]0))
56	22, 55	eldifd 3909	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (ℂ ∖ (-∞(,]0)))
57	56	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ (ℂ ∖ (-∞(,]0)))
58		ovexd 7387	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))) ∈ V)
59		eldifi 4080	. . . . . . . 8 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
60		eldifn 4081	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
61		0xr 11166	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
62		mnflt0 13026	. . . . . . . . . . . 12 ⊢ -∞ < 0
63		ubioc1 13301	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
64	51, 61, 62, 63	mp3an 1463	. . . . . . . . . . 11 ⊢ 0 ∈ (-∞(,]0)
65		eleq1 2821	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
66	64, 65	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
67	66	necon3bi 2955	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (-∞(,]0) → 𝑦 ≠ 0)
68	60, 67	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ≠ 0)
69	59, 68	logcld 26507	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → (log‘𝑦) ∈ ℂ)
70	69	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
71		ovexd 7387	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
72	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → i ∈ ℂ)
73	72, 12	mulcld 11139	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (i · 𝑥) ∈ ℂ)
74	73	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (i · 𝑥) ∈ ℂ)
75	13	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → i ∈ ℂ)
76	12	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
77		1cnd 11114	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 1 ∈ ℂ)
78		simpr 484	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
79		1cnd 11114	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
80	11	dvmptid 25889	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
81	5	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 𝐷 ⊆ ℂ)
82		eqid 2733	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
83	82	cnfldtopon 24698	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
84	83	toponrestid 22837	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
85	82	recld2 24731	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂ_fld))
86		neg1rr 12118	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℝ
87		iocmnfcld 24684	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ∈ ℝ → (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))))
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,)))
89		1re 11119	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
90		icopnfcld 24683	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
91	89, 90	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))
92		uncld 22957	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))) ∧ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,))))
93	88, 91, 92	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,)))
94		tgioo4 24721	. . . . . . . . . . . . . . . . 17 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
95	94	fveq2i 6831	. . . . . . . . . . . . . . . 16 ⊢ (Clsd‘(topGen‘ran (,))) = (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
96	93, 95	eleqtri 2831	. . . . . . . . . . . . . . 15 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t ℝ))
97		restcldr 23090	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . 14 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂ_fld))
99	83	toponunii 22832	. . . . . . . . . . . . . . 15 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
100	99	cldopn 22947	. . . . . . . . . . . . . 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 2829	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
103	102	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
104	11, 78, 79, 80, 81, 84, 82, 103	dvmptres 25895	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ 1))
105	13	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → i ∈ ℂ)
106	11, 76, 77, 104, 105	dvmptcmul 25896	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (i · 𝑥))) = (𝑥 ∈ 𝐷 ↦ (i · 1)))
107	13	mulridi 11123	. . . . . . . . . 10 ⊢ (i · 1) = i
108	107	mpteq2i 5189	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↦ (i · 1)) = (𝑥 ∈ 𝐷 ↦ i)
109	106, 108	eqtrdi 2784	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (i · 𝑥))) = (𝑥 ∈ 𝐷 ↦ i))
110	12	sqcld 14053	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑2) ∈ ℂ)
111	16, 110, 18	sylancr 587	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (1 − (𝑥↑2)) ∈ ℂ)
112	111	sqrtcld 15349	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (√‘(1 − (𝑥↑2))) ∈ ℂ)
113	112	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘(1 − (𝑥↑2))) ∈ ℂ)
114		ovexd 7387	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (-𝑥 / (√‘(1 − (𝑥↑2)))) ∈ V)
115		elin 3914	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐷 ∩ ℝ) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ))
116	3	asindmre 37764	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)
117	116	eqimssi 3991	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∩ ℝ) ⊆ (-1(,)1)
118	117	sseli 3926	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐷 ∩ ℝ) → 𝑥 ∈ (-1(,)1))
119	115, 118	sylbir 235	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ (-1(,)1))
120		incom 4158	. . . . . . . . . . . . . . . 16 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ((-∞(,]0) ∩ (0(,)+∞))
121		pnfxr 11173	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
122		df-ioc 13252	. . . . . . . . . . . . . . . . . 18 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
123		df-ioo 13251	. . . . . . . . . . . . . . . . . 18 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
124		xrltnle 11186	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (0 < 𝑤 ↔ ¬ 𝑤 ≤ 0))
125	122, 123, 124	ixxdisj 13262	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
126	51, 61, 121, 125	mp3an 1463	. . . . . . . . . . . . . . . 16 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
127	120, 126	eqtri 2756	. . . . . . . . . . . . . . 15 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
128		elioore 13277	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (-1(,)1) → 𝑥 ∈ ℝ)
129	128	resqcld 14034	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (-1(,)1) → (𝑥↑2) ∈ ℝ)
130		resubcl 11432	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (1 − (𝑥↑2)) ∈ ℝ)
131	89, 129, 130	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ ℝ)
132	86	rexri 11177	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ^*
133		1xr 11178	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
134		elioo2 13288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → (𝑥 ∈ (-1(,)1) ↔ (𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1)))
135	132, 133, 134	mp2an 692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (-1(,)1) ↔ (𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1))
136		recn 11103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
137	136	abscld 15348	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
138	136	absge0d 15356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → 0 ≤ (abs‘𝑥))
139		0le1 11647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ≤ 1
140		lt2sq 14042	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (1 ∈ ℝ ∧ 0 ≤ 1)) → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
141	89, 139, 140	mpanr12 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
142	137, 138, 141	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥) < 1 ↔ ((abs‘𝑥)↑2) < (1↑2)))
143		abslt 15224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑥) < 1 ↔ (-1 < 𝑥 ∧ 𝑥 < 1)))
144	89, 143	mpan2 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥) < 1 ↔ (-1 < 𝑥 ∧ 𝑥 < 1)))
145		absresq 15211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → ((abs‘𝑥)↑2) = (𝑥↑2))
146		sq1 14104	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1↑2) = 1
147	146	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → (1↑2) = 1)
148	145, 147	breq12d 5106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (((abs‘𝑥)↑2) < (1↑2) ↔ (𝑥↑2) < 1))
149		resqcl 14033	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → (𝑥↑2) ∈ ℝ)
150		posdif 11617	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥↑2) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑥↑2) < 1 ↔ 0 < (1 − (𝑥↑2))))
151	149, 89, 150	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → ((𝑥↑2) < 1 ↔ 0 < (1 − (𝑥↑2))))
152	148, 151	bitrd 279	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → (((abs‘𝑥)↑2) < (1↑2) ↔ 0 < (1 − (𝑥↑2))))
153	142, 144, 152	3bitr3d 309	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ → ((-1 < 𝑥 ∧ 𝑥 < 1) ↔ 0 < (1 − (𝑥↑2))))
154	153	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → ((-1 < 𝑥 ∧ 𝑥 < 1) → 0 < (1 − (𝑥↑2))))
155	154	3impib 1116	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ -1 < 𝑥 ∧ 𝑥 < 1) → 0 < (1 − (𝑥↑2)))
156	135, 155	sylbi 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-1(,)1) → 0 < (1 − (𝑥↑2)))
157	131, 156	elrpd 12933	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ ℝ⁺)
158		ioorp 13327	. . . . . . . . . . . . . . . 16 ⊢ (0(,)+∞) = ℝ⁺
159	157, 158	eleqtrrdi 2844	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (-1(,)1) → (1 − (𝑥↑2)) ∈ (0(,)+∞))
160		disjel 4406	. . . . . . . . . . . . . . 15 ⊢ ((((0(,)+∞) ∩ (-∞(,]0)) = ∅ ∧ (1 − (𝑥↑2)) ∈ (0(,)+∞)) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
161	127, 159, 160	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (-1(,)1) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
162	119, 161	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ ℝ) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
163		elioc2 13311	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → ((1 − (𝑥↑2)) ∈ (-∞(,]0) ↔ ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0)))
164	51, 36, 163	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) ↔ ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0))
165	164	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → ((1 − (𝑥↑2)) ∈ ℝ ∧ -∞ < (1 − (𝑥↑2)) ∧ (1 − (𝑥↑2)) ≤ 0))
166	165	simp1d 1142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (𝑥↑2)) ∈ ℝ)
167		resubcl 11432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ ∧ (1 − (𝑥↑2)) ∈ ℝ) → (1 − (1 − (𝑥↑2))) ∈ ℝ)
168	89, 166, 167	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (1 − (𝑥↑2))) ∈ ℝ)
169		nncan 11397	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
170	16, 169	mpan 690	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥↑2) ∈ ℂ → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
171	170	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥↑2) ∈ ℂ → ((1 − (1 − (𝑥↑2))) ∈ ℝ ↔ (𝑥↑2) ∈ ℝ))
172	171	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (1 − (𝑥↑2))) ∈ ℝ) → (𝑥↑2) ∈ ℝ)
173	168, 172	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥↑2) ∈ ℝ)
174	166	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (𝑥↑2)) ∈ ℝ)
175	165	simp3d 1144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ (-∞(,]0) → (1 − (𝑥↑2)) ≤ 0)
176	175	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (𝑥↑2)) ≤ 0)
177		letr 11214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 − (𝑥↑2)) ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ) → (((1 − (𝑥↑2)) ≤ 0 ∧ 0 ≤ 1) → (1 − (𝑥↑2)) ≤ 1))
178	36, 89, 177	mp3an23 1455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 − (𝑥↑2)) ∈ ℝ → (((1 − (𝑥↑2)) ≤ 0 ∧ 0 ≤ 1) → (1 − (𝑥↑2)) ≤ 1))
179	139, 178	mpan2i 697	. . . . . . . . . . . . . . . . . . . . . 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 11637	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℝ ∧ (1 − (𝑥↑2)) ∈ ℝ) → (0 ≤ (1 − (1 − (𝑥↑2))) ↔ (1 − (𝑥↑2)) ≤ 1))
182	89, 174, 181	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (0 ≤ (1 − (1 − (𝑥↑2))) ↔ (1 − (𝑥↑2)) ≤ 1))
183	180, 182	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 0 ≤ (1 − (1 − (𝑥↑2))))
184	170	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (1 − (1 − (𝑥↑2))) = (𝑥↑2))
185	183, 184	breqtrd 5119	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 0 ≤ (𝑥↑2))
186	173, 185	resqrtcld 15327	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥↑2) ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (√‘(𝑥↑2)) ∈ ℝ)
187	17, 186	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (√‘(𝑥↑2)) ∈ ℝ)
188		eleq1 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (√‘(𝑥↑2)) → (𝑥 ∈ ℝ ↔ (√‘(𝑥↑2)) ∈ ℝ))
189	187, 188	syl5ibrcom 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = (√‘(𝑥↑2)) → 𝑥 ∈ ℝ))
190	187	renegcld 11551	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → -(√‘(𝑥↑2)) ∈ ℝ)
191		eleq1 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -(√‘(𝑥↑2)) → (𝑥 ∈ ℝ ↔ -(√‘(𝑥↑2)) ∈ ℝ))
192	190, 191	syl5ibrcom 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = -(√‘(𝑥↑2)) → 𝑥 ∈ ℝ))
193		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥↑2) = (𝑥↑2)
194		eqsqrtor 15276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → ((𝑥↑2) = (𝑥↑2) ↔ (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2)))))
195	17, 194	mpdan 687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = (𝑥↑2) ↔ (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2)))))
196	193, 195	mpbii 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2))))
197	196	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → (𝑥 = (√‘(𝑥↑2)) ∨ 𝑥 = -(√‘(𝑥↑2))))
198	189, 192, 197	mpjaod 860	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ (1 − (𝑥↑2)) ∈ (-∞(,]0)) → 𝑥 ∈ ℝ)
199	198	stoic1a 1773	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ ¬ 𝑥 ∈ ℝ) → ¬ (1 − (𝑥↑2)) ∈ (-∞(,]0))
200	12, 199	sylan 580	. . . . . . . . . . . . 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 3909	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (1 − (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0)))
203	202	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (1 − (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0)))
204		2cnd 12210	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 2 ∈ ℂ)
205		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
206	204, 205	mulcld 11139	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (2 · 𝑥) ∈ ℂ)
207	206	negcld 11466	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → -(2 · 𝑥) ∈ ℂ)
208	207	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(2 · 𝑥) ∈ ℂ)
209	12, 208	sylan2 593	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → -(2 · 𝑥) ∈ ℂ)
210	59	sqrtcld 15349	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → (√‘𝑦) ∈ ℂ)
211	210	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (√‘𝑦) ∈ ℂ)
212		ovexd 7387	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / (2 · (√‘𝑦))) ∈ V)
213	19	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1 − (𝑥↑2)) ∈ ℂ)
214	36	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 0 ∈ ℝ)
215		1cnd 11114	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℂ)
216	11, 215	dvmptc 25890	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ 1)) = (𝑥 ∈ ℂ ↦ 0))
217	17	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥↑2) ∈ ℂ)
218		2cn 12207	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
219		mulcl 11097	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
220	218, 219	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (2 · 𝑥) ∈ ℂ)
221	220	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
222		2nn 12205	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
223		dvexp 25885	. . . . . . . . . . . . . . . 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 12253	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 − 1) = 1
226	225	oveq2i 7363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥↑(2 − 1)) = (𝑥↑1)
227		exp1 13976	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥↑1) = 𝑥)
228	226, 227	eqtrid 2780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑥↑(2 − 1)) = 𝑥)
229	228	oveq2d 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (2 · (𝑥↑(2 − 1))) = (2 · 𝑥))
230	229	mpteq2ia 5188	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ ↦ (2 · (𝑥↑(2 − 1)))) = (𝑥 ∈ ℂ ↦ (2 · 𝑥))
231	224, 230	eqtri 2756	. . . . . . . . . . . . . 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 25899	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ ℂ ↦ (0 − (2 · 𝑥))))
234		df-neg 11354	. . . . . . . . . . . . 13 ⊢ -(2 · 𝑥) = (0 − (2 · 𝑥))
235	234	mpteq2i 5189	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(2 · 𝑥)) = (𝑥 ∈ ℂ ↦ (0 − (2 · 𝑥)))
236	233, 235	eqtr4di 2786	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ ℂ ↦ -(2 · 𝑥)))
237	11, 213, 208, 236, 81, 84, 82, 103	dvmptres 25895	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (1 − (𝑥↑2)))) = (𝑥 ∈ 𝐷 ↦ -(2 · 𝑥)))
238		eqid 2733	. . . . . . . . . . . 12 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
239	238	dvcnsqrt 26681	. . . . . . . . . . 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 6828	. . . . . . . . . 10 ⊢ (𝑦 = (1 − (𝑥↑2)) → (√‘𝑦) = (√‘(1 − (𝑥↑2))))
242	241	oveq2d 7368	. . . . . . . . . . 11 ⊢ (𝑦 = (1 − (𝑥↑2)) → (2 · (√‘𝑦)) = (2 · (√‘(1 − (𝑥↑2)))))
243	242	oveq2d 7368	. . . . . . . . . 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 25904	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘(1 − (𝑥↑2))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥))))
245		mulneg2 11561	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · -𝑥) = -(2 · 𝑥))
246	218, 12, 245	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · -𝑥) = -(2 · 𝑥))
247	246	oveq1d 7367	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-(2 · 𝑥) / (2 · (√‘(1 − (𝑥↑2))))))
248	12	negcld 11466	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → -𝑥 ∈ ℂ)
249		eldifn 4081	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
250	249, 3	eleq2s 2851	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 → ¬ 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
251		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = -1 → 𝑥 = -1)
252		mnflt 13024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-1 ∈ ℝ → -∞ < -1)
253	86, 252	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ -∞ < -1
254		ubioc1 13301	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ -∞ < -1) → -1 ∈ (-∞(,]-1))
255	51, 132, 253, 254	mp3an 1463	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ (-∞(,]-1)
256	251, 255	eqeltrdi 2841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = -1 → 𝑥 ∈ (-∞(,]-1))
257		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 1 → 𝑥 = 1)
258		ltpnf 13021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ ℝ → 1 < +∞)
259	89, 258	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < +∞
260		lbico1 13302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 1 < +∞) → 1 ∈ (1[,)+∞))
261	133, 121, 259, 260	mp3an 1463	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ (1[,)+∞)
262	257, 261	eqeltrdi 2841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → 𝑥 ∈ (1[,)+∞))
263	256, 262	orim12i 908	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = -1 ∨ 𝑥 = 1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
264	263	orcoms 872	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
265		elun 4102	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)) ↔ (𝑥 ∈ (-∞(,]-1) ∨ 𝑥 ∈ (1[,)+∞)))
266	264, 265	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ((-∞(,]-1) ∪ (1[,)+∞)))
267	250, 266	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → ¬ (𝑥 = 1 ∨ 𝑥 = -1))
268		1cnd 11114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 ∈ ℂ)
269	17	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) ∈ ℂ)
270	19	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) ∈ ℂ)
271		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (√‘(1 − (𝑥↑2))) = 0)
272	270, 271	sqr00d 15353	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (1 − (𝑥↑2)) = 0)
273	268, 269, 272	subeq0d 11487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → 1 = (𝑥↑2))
274	146, 273	eqtr2id 2781	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) = 0) → (𝑥↑2) = (1↑2))
275	274	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥↑2) = (1↑2)))
276		sqeqor 14125	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
277	16, 276	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) = (1↑2) ↔ (𝑥 = 1 ∨ 𝑥 = -1)))
278	275, 277	sylibd 239	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((√‘(1 − (𝑥↑2))) = 0 → (𝑥 = 1 ∨ 𝑥 = -1)))
279	278	necon3bd 2943	. . . . . . . . . . . . 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 12337	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
282		divcan5 11830	. . . . . . . . . . . . 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 1452	. . . . . . . . . . . 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 836	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → ((2 · -𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
285	218, 12, 219	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → (2 · 𝑥) ∈ ℂ)
286	285	negcld 11466	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → -(2 · 𝑥) ∈ ℂ)
287		mulcl 11097	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ∈ ℂ) → (2 · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
288	218, 112, 287	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
289		mulne0 11766	. . . . . . . . . . . . . 14 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0)) → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
290	281, 289	mpan 690	. . . . . . . . . . . . 13 ⊢ (((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0) → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
291	112, 280, 290	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → (2 · (√‘(1 − (𝑥↑2)))) ≠ 0)
292	286, 288, 291	divrec2d 11908	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-(2 · 𝑥) / (2 · (√‘(1 − (𝑥↑2))))) = ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥)))
293	247, 284, 292	3eqtr3rd 2777	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥)) = (-𝑥 / (√‘(1 − (𝑥↑2)))))
294	293	mpteq2ia 5188	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (2 · (√‘(1 − (𝑥↑2))))) · -(2 · 𝑥))) = (𝑥 ∈ 𝐷 ↦ (-𝑥 / (√‘(1 − (𝑥↑2)))))
295	244, 294	eqtrdi 2784	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘(1 − (𝑥↑2))))) = (𝑥 ∈ 𝐷 ↦ (-𝑥 / (√‘(1 − (𝑥↑2))))))
296	11, 74, 75, 109, 113, 114, 295	dvmptadd 25892	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (i + (-𝑥 / (√‘(1 − (𝑥↑2)))))))
297		mulcl 11097	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ∈ ℂ) → (i · (√‘(1 − (𝑥↑2)))) ∈ ℂ)
298	13, 20, 297	sylancr 587	. . . . . . . . . . 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 11942	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((i · (√‘(1 − (𝑥↑2)))) + -𝑥) / (√‘(1 − (𝑥↑2)))) = (((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) + (-𝑥 / (√‘(1 − (𝑥↑2))))))
301		ixi 11753	. . . . . . . . . . . . . . . 16 ⊢ (i · i) = -1
302	301	eqcomi 2742	. . . . . . . . . . . . . . 15 ⊢ -1 = (i · i)
303	302	oveq1i 7362	. . . . . . . . . . . . . 14 ⊢ (-1 · 𝑥) = ((i · i) · 𝑥)
304		mulm1 11565	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (-1 · 𝑥) = -𝑥)
305		mulass 11101	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((i · i) · 𝑥) = (i · (i · 𝑥)))
306	13, 13, 305	mp3an12 1453	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((i · i) · 𝑥) = (i · (i · 𝑥)))
307	303, 304, 306	3eqtr3a 2792	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → -𝑥 = (i · (i · 𝑥)))
308	307	oveq1d 7367	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (-𝑥 + (i · (√‘(1 − (𝑥↑2))))) = ((i · (i · 𝑥)) + (i · (√‘(1 − (𝑥↑2))))))
309		negcl 11367	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ)
310	298, 309	addcomd 11322	. . . . . . . . . . . 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 11143	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = ((i · (i · 𝑥)) + (i · (√‘(1 − (𝑥↑2))))))
313	308, 310, 312	3eqtr4d 2778	. . . . . . . . . . 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 7367	. . . . . . . . 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 11910	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) = i)
317	316	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((i · (√‘(1 − (𝑥↑2)))) / (√‘(1 − (𝑥↑2)))) + (-𝑥 / (√‘(1 − (𝑥↑2))))) = (i + (-𝑥 / (√‘(1 − (𝑥↑2))))))
318	300, 315, 317	3eqtr3rd 2777	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (i + (-𝑥 / (√‘(1 − (𝑥↑2))))) = ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))
319	318	mpteq2ia 5188	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↦ (i + (-𝑥 / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2)))))
320	296, 319	eqtrdi 2784	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / (√‘(1 − (𝑥↑2))))))
321		logf1o 26501	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
322		f1of 6768	. . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
323	321, 322	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → log:(ℂ ∖ {0})⟶ran log)
324		snssi 4759	. . . . . . . . . . 11 ⊢ (0 ∈ (-∞(,]0) → {0} ⊆ (-∞(,]0))
325	64, 324	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ (-∞(,]0)
326		sscon 4092	. . . . . . . . . 10 ⊢ ({0} ⊆ (-∞(,]0) → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
327	325, 326	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
328	323, 327	feqresmpt 6897	. . . . . . . 8 ⊢ (⊤ → (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
329	328	oveq2d 7368	. . . . . . 7 ⊢ (⊤ → (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))))
330	238	dvlog 26588	. . . . . . 7 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
331	329, 330	eqtr3di 2783	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦)))
332		fveq2 6828	. . . . . 6 ⊢ (𝑦 = ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) → (log‘𝑦) = (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))
333		oveq2 7360	. . . . . 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 25904	. . . . 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 11897	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
336		mulcl 11097	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) ∈ ℂ) → (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) ∈ ℂ)
337	13, 21, 336	sylancr 587	. . . . . . . . 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 11939	. . . . . . 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 11908	. . . . . . . . 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 11910	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ((i · ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = i)
342	340, 341	eqtr3d 2770	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((1 / ((i · 𝑥) + (√‘(1 − (𝑥↑2))))) · (i · ((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = i)
343	342	oveq1d 7367	. . . . . . 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 2770	. . . . . 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 5188	. . . . 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 2784	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) = (𝑥 ∈ 𝐷 ↦ (i / (√‘(1 − (𝑥↑2))))))
347		negicn 11368	. . . . 5 ⊢ -i ∈ ℂ
348	347	a1i 11	. . . 4 ⊢ (⊤ → -i ∈ ℂ)
349	11, 26, 27, 346, 348	dvmptcmul 25896	. . 3 ⊢ (⊤ → (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))) = (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2)))))))
350	349	mptru 1548	. 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))) = (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2))))))
351		divass 11801	. . . . . 6 ⊢ ((-i ∈ ℂ ∧ i ∈ ℂ ∧ ((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0)) → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
352	347, 13, 351	mp3an12 1453	. . . . 5 ⊢ (((√‘(1 − (𝑥↑2))) ∈ ℂ ∧ (√‘(1 − (𝑥↑2))) ≠ 0) → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
353	112, 280, 352	syl2anc 584	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((-i · i) / (√‘(1 − (𝑥↑2)))) = (-i · (i / (√‘(1 − (𝑥↑2))))))
354	13, 13	mulneg1i 11570	. . . . . 6 ⊢ (-i · i) = -(i · i)
355	301	negeqi 11360	. . . . . 6 ⊢ -(i · i) = --1
356		negneg1e1 12121	. . . . . 6 ⊢ --1 = 1
357	354, 355, 356	3eqtri 2760	. . . . 5 ⊢ (-i · i) = 1
358	357	oveq1i 7362	. . . 4 ⊢ ((-i · i) / (√‘(1 − (𝑥↑2)))) = (1 / (√‘(1 − (𝑥↑2))))
359	353, 358	eqtr3di 2783	. . 3 ⊢ (𝑥 ∈ 𝐷 → (-i · (i / (√‘(1 − (𝑥↑2))))) = (1 / (√‘(1 − (𝑥↑2)))))
360	359	mpteq2ia 5188	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (-i · (i / (√‘(1 − (𝑥↑2)))))) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))
361	9, 350, 360	3eqtri 2760	1 ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2929 Vcvv 3437 ∖ cdif 3895 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 {csn 4575 {cpr 4577 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 ↾ cres 5621 ⟶wf 6482 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 ici 11015 + caddc 11016 · cmul 11018 +∞cpnf 11150 -∞cmnf 11151 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154 − cmin 11351 -cneg 11352 / cdiv 11781 ℕcn 12132 2c2 12187 ℝ⁺crp 12892 (,)cioo 13247 (,]cioc 13248 [,)cico 13249 ↑cexp 13970 ℜcre 15006 √csqrt 15142 abscabs 15143 ↾_t crest 17326 TopOpenctopn 17327 topGenctg 17343 ℂ_fldccnfld 21293 Clsdccld 22932 D cdv 25792 logclog 26491 arcsincasin 26800

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ioc 13252 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-fl 13698 df-mod 13776 df-seq 13911 df-exp 13971 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14976 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-limsup 15380 df-clim 15397 df-rlim 15398 df-sum 15596 df-ef 15976 df-sin 15978 df-cos 15979 df-tan 15980 df-pi 15981 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796 df-log 26493 df-cxp 26494 df-asin 26803

This theorem is referenced by: dvacos 37766 dvreasin 37767

W3C validator