asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 38411

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 4126	. . . . 5 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞)))
2		neg1rr 12219	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 11282	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 11283	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 11278	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1358	. . . . . . . 8 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 12221	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11751	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 11225	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 11223	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 11351	. . . . . . . . . 10 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 705	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 13161	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
14	10, 13	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
15	12, 14	pm3.2i 476	. . . . . . . 8 ⊢ (-1 < 1 ∧ 1 < +∞)
16		df-ioo 13392	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 13394	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 11289	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 13181	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 13198	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 13404	. . . . . . . 8 ⊢ (((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-1 < 1 ∧ 1 < +∞)) → ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞))
22	6, 15, 21	mp2an 705	. . . . . . 7 ⊢ ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞)
23	22	uneq2i 4119	. . . . . 6 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ (-1(,)+∞))
24		mnfxr 11281	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1358	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 13164	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -∞ < -1)
27		ltpnf 13161	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -1 < +∞)
28	26, 27	jca 521	. . . . . . . 8 ⊢ (-1 ∈ ℝ → (-∞ < -1 ∧ -1 < +∞))
29	2, 28	ax-mp 5	. . . . . . 7 ⊢ (-∞ < -1 ∧ -1 < +∞)
30		df-ioc 13393	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 11291	. . . . . . . 8 ⊢ ((-1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 13197	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 13181	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 13404	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 705	. . . . . 6 ⊢ ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2788	. . . . 5 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 13465	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2792	. . . 4 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4077	. . 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 11172	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4254	. . . 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 2797	. 2 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
46	45	ineq1i 4169	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4162	. . . . 5 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ((-∞(,]-1) ∩ (-1(,)1))
48	30, 16, 31	ixxdisj 13403	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,]-1) ∩ (-1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1490	. . . . 5 ⊢ ((-∞(,]-1) ∩ (-1(,)1)) = ∅
50	47, 49	eqtri 2788	. . . 4 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ∅
51	16, 17, 18	ixxdisj 13403	. . . . 5 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1490	. . . 4 ⊢ ((-1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 476	. . 3 ⊢ (((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4364	. . . 4 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4237	. . . . 5 ⊢ ((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2770	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4414	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
58	54, 56, 57	3bitr2i 302	. . 3 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞))))
59	53, 58	mpbi 233	. 2 ⊢ (-1(,)1) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
60	44, 46, 59	3eqtr4i 2798	1 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 class class class wbr 5111 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 +∞cpnf 11255 -∞cmnf 11256 ℝ^*cxr 11257 < clt 11258 ≤ cle 11259 -cneg 11457 (,)cioo 13388 (,]cioc 13389 [,)cico 13390

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-ioo 13392 df-ioc 13393 df-ico 13394

This theorem is used by: dvasin 38412 dvreasin 38414 dvreacos 38415

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