asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 37663

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 4196	. . . . 5 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞)))
2		neg1rr 12408	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 11348	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 11349	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 11344	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1339	. . . . . . . 8 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 12410	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11812	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 11292	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 11290	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 11416	. . . . . . . . . 10 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 691	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 13183	. . . . . . . . . 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 13411	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 13413	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 11355	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 13202	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 13219	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 13423	. . . . . . . 8 ⊢ (((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-1 < 1 ∧ 1 < +∞)) → ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞))
22	6, 15, 21	mp2an 691	. . . . . . 7 ⊢ ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞)
23	22	uneq2i 4188	. . . . . 6 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ (-1(,)+∞))
24		mnfxr 11347	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1339	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 13186	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -∞ < -1)
27		ltpnf 13183	. . . . . . . . 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 13412	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 11357	. . . . . . . 8 ⊢ ((-1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 13218	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 13202	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 13423	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 691	. . . . . 6 ⊢ ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2768	. . . . 5 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 13482	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2772	. . . 4 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4145	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
40		difun2 4504	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
41		ax-resscn 11241	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4320	. . . 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 2777	. 2 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
46	45	ineq1i 4237	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4230	. . . . 5 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ((-∞(,]-1) ∩ (-1(,)1))
48	30, 16, 31	ixxdisj 13422	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,]-1) ∩ (-1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1461	. . . . 5 ⊢ ((-∞(,]-1) ∩ (-1(,)1)) = ∅
50	47, 49	eqtri 2768	. . . 4 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ∅
51	16, 17, 18	ixxdisj 13422	. . . . 5 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1461	. . . 4 ⊢ ((-1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 470	. . 3 ⊢ (((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4468	. . . 4 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4303	. . . . 5 ⊢ ((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2745	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4477	. . . 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 2778	1 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 -cneg 11521 (,)cioo 13407 (,]cioc 13408 [,)cico 13409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-ioo 13411 df-ioc 13412 df-ico 13413

This theorem is referenced by: dvasin 37664 dvreasin 37666 dvreacos 37667

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