asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 37697

Description: Real part of domain of differentiability of arcsine. (Contributed by Brendan Leahy, 19-Dec-2018.)

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

**Assertion**
Ref	Expression
asindmre	⊢ (𝐷 ∩ ℝ) = (-1(,)1)

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

Step	Hyp	Ref	Expression
1		un12 4136	. . . . 5 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞)))
2		neg1rr 12172	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 11232	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 11233	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 11228	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1340	. . . . . . . 8 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 12174	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11700	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 11176	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 11174	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 11300	. . . . . . . . . 10 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 692	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 13080	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
14	10, 13	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
15	12, 14	pm3.2i 470	. . . . . . . 8 ⊢ (-1 < 1 ∧ 1 < +∞)
16		df-ioo 13310	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 13312	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 11239	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 13100	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 13117	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 13322	. . . . . . . 8 ⊢ (((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-1 < 1 ∧ 1 < +∞)) → ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞))
22	6, 15, 21	mp2an 692	. . . . . . 7 ⊢ ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞)
23	22	uneq2i 4128	. . . . . 6 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ (-1(,)+∞))
24		mnfxr 11231	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1340	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 13083	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -∞ < -1)
27		ltpnf 13080	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -1 < +∞)
28	26, 27	jca 511	. . . . . . . 8 ⊢ (-1 ∈ ℝ → (-∞ < -1 ∧ -1 < +∞))
29	2, 28	ax-mp 5	. . . . . . 7 ⊢ (-∞ < -1 ∧ -1 < +∞)
30		df-ioc 13311	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 11241	. . . . . . . 8 ⊢ ((-1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 13116	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 13100	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 13322	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 692	. . . . . 6 ⊢ ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2752	. . . . 5 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 13383	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2756	. . . 4 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4085	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
40		difun2 4444	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
41		ax-resscn 11125	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4264	. . . 4 ⊢ (ℝ ⊆ ℂ → (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ))
43	41, 42	ax-mp 5	. . 3 ⊢ (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
44	39, 40, 43	3eqtr3ri 2761	. 2 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
46	45	ineq1i 4179	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4172	. . . . 5 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ((-∞(,]-1) ∩ (-1(,)1))
48	30, 16, 31	ixxdisj 13321	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,]-1) ∩ (-1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1463	. . . . 5 ⊢ ((-∞(,]-1) ∩ (-1(,)1)) = ∅
50	47, 49	eqtri 2752	. . . 4 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ∅
51	16, 17, 18	ixxdisj 13321	. . . . 5 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1463	. . . 4 ⊢ ((-1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 470	. . 3 ⊢ (((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4408	. . . 4 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4247	. . . . 5 ⊢ ((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2734	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4417	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
58	54, 56, 57	3bitr2i 299	. . 3 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
59	53, 58	mpbi 230	. 2 ⊢ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
60	44, 46, 59	3eqtr4i 2762	1 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 class class class wbr 5107 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 -cneg 11406 (,)cioo 13306 (,]cioc 13307 [,)cico 13308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-ioo 13310 df-ioc 13311 df-ico 13312

This theorem is referenced by: dvasin 37698 dvreasin 37700 dvreacos 37701

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