asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 34979

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 4145	. . . . 5 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞)))
2		neg1rr 11755	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 10701	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 10702	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 10697	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1335	. . . . . . . 8 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 11757	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11164	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 10645	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 10643	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 10768	. . . . . . . . . 10 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 690	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 12518	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
14	10, 13	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
15	12, 14	pm3.2i 473	. . . . . . . 8 ⊢ (-1 < 1 ∧ 1 < +∞)
16		df-ioo 12745	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 12747	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 10708	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 12536	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 12553	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 12757	. . . . . . . 8 ⊢ (((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-1 < 1 ∧ 1 < +∞)) → ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞))
22	6, 15, 21	mp2an 690	. . . . . . 7 ⊢ ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞)
23	22	uneq2i 4138	. . . . . 6 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ (-1(,)+∞))
24		mnfxr 10700	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1335	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 12521	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -∞ < -1)
27		ltpnf 12518	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -1 < +∞)
28	26, 27	jca 514	. . . . . . . 8 ⊢ (-1 ∈ ℝ → (-∞ < -1 ∧ -1 < +∞))
29	2, 28	ax-mp 5	. . . . . . 7 ⊢ (-∞ < -1 ∧ -1 < +∞)
30		df-ioc 12746	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 10710	. . . . . . . 8 ⊢ ((-1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 12552	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 12536	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 12757	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 690	. . . . . 6 ⊢ ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2846	. . . . 5 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 12814	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2850	. . . 4 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4097	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
40		difun2 4431	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
41		ax-resscn 10596	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4268	. . . 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 2855	. 2 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
46	45	ineq1i 4187	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4180	. . . . 5 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ((-∞(,]-1) ∩ (-1(,)1))
48	30, 16, 31	ixxdisj 12756	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,]-1) ∩ (-1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1457	. . . . 5 ⊢ ((-∞(,]-1) ∩ (-1(,)1)) = ∅
50	47, 49	eqtri 2846	. . . 4 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ∅
51	16, 17, 18	ixxdisj 12756	. . . . 5 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1457	. . . 4 ⊢ ((-1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 473	. . 3 ⊢ (((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4396	. . . 4 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4252	. . . . 5 ⊢ ((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2828	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4405	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
58	54, 56, 57	3bitr2i 301	. . 3 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
59	53, 58	mpbi 232	. 2 ⊢ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
60	44, 46, 59	3eqtr4i 2856	1 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 +∞cpnf 10674 -∞cmnf 10675 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 -cneg 10873 (,)cioo 12741 (,]cioc 12742 [,)cico 12743

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-ioo 12745 df-ioc 12746 df-ico 12747

This theorem is referenced by: dvasin 34980 dvreasin 34982 dvreacos 34983

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