asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 38889

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 4174	. . . . 5 ⊢ (( -1(,)1) ∪ ((-∞(,] -1) ∪ (1[,)+∞))) = ((-∞(,] -1) ∪ (( -1(,)1) ∪ (1[,)+∞)))
2		neg1rr 12346	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 11406	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 11407	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 11402	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1358	. . . . . . . 8 ⊢ ( -1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 12348	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11878	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 11349	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 11347	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 11475	. . . . . . . . . 10 ⊢ (( -1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 705	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 13289	. . . . . . . . . 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 13520	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 13522	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 11413	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 13309	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 13326	. . . . . . . . 9 ⊢ (( -1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (( -1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 13532	. . . . . . . 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 4167	. . . . . 6 ⊢ ((-∞(,] -1) ∪ (( -1(,)1) ∪ (1[,)+∞))) = ((-∞(,] -1) ∪ ( -1(,)+∞))
24		mnfxr 11405	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1358	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 13292	. . . . . . . . 9 ⊢ ( -1 ∈ ℝ → -∞ < -1)
27		ltpnf 13289	. . . . . . . . 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 13521	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 11415	. . . . . . . 8 ⊢ (( -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ( -1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 13325	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 13309	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 13532	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,] -1) ∪ ( -1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 705	. . . . . 6 ⊢ ((-∞(,] -1) ∪ ( -1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2786	. . . . 5 ⊢ ((-∞(,] -1) ∪ (( -1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 13593	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2790	. . . 4 ⊢ (( -1(,)1) ∪ ((-∞(,] -1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4062	. . 3 ⊢ ((( -1(,)1) ∪ ((-∞(,] -1) ∪ (1[,)+∞))) ∖ ((-∞(,] -1) ∪ (1[,)+∞))) = (ℝ ∖ ((-∞(,] -1) ∪ (1[,)+∞)))
40		difun2 4440	. . 3 ⊢ ((( -1(,)1) ∪ ((-∞(,] -1) ∪ (1[,)+∞))) ∖ ((-∞(,] -1) ∪ (1[,)+∞))) = (( -1(,)1) ∖ ((-∞(,] -1) ∪ (1[,)+∞)))
41		ax-resscn 11296	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4249	. . . 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 2795	. 2 ⊢ ((ℂ ∖ ((-∞(,] -1) ∪ (1[,)+∞))) ∩ ℝ) = (( -1(,)1) ∖ ((-∞(,] -1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,] -1) ∪ (1[,)+∞)))
46	45	ineq1i 4116	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,] -1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4109	. . . . 5 ⊢ (( -1(,)1) ∩ (-∞(,] -1)) = ((-∞(,] -1) ∩ ( -1(,)1))
48	30, 16, 31	ixxdisj 13531	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,] -1) ∩ ( -1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1491	. . . . 5 ⊢ ((-∞(,] -1) ∩ ( -1(,)1)) = ∅
50	47, 49	eqtri 2786	. . . 4 ⊢ (( -1(,)1) ∩ (-∞(,] -1)) = ∅
51	16, 17, 18	ixxdisj 13531	. . . . 5 ⊢ (( -1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (( -1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1491	. . . 4 ⊢ (( -1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 476	. . 3 ⊢ ((( -1(,)1) ∩ (-∞(,] -1)) = ∅ ∧ (( -1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4360	. . . 4 ⊢ (((( -1(,)1) ∩ (-∞(,] -1)) = ∅ ∧ (( -1(,)1) ∩ (1[,)+∞)) = ∅) ↔ ((( -1(,)1) ∩ (-∞(,] -1)) ∪ (( -1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4232	. . . . 5 ⊢ (( -1(,)1) ∩ ((-∞(,] -1) ∪ (1[,)+∞))) = ((( -1(,)1) ∩ (-∞(,] -1)) ∪ (( -1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2768	. . . 4 ⊢ ((( -1(,)1) ∩ ((-∞(,] -1) ∪ (1[,)+∞))) = ∅ ↔ ((( -1(,)1) ∩ (-∞(,] -1)) ∪ (( -1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4410	. . . 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 2796	1 ⊢ (𝐷 ∩ ℝ) = ( -1(,)1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 class class class wbr 5108 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 -cneg 11581 (,)cioo 13516 (,]cioc 13517 [,)cico 13518

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-ioo 13520 df-ioc 13521 df-ico 13522

This theorem is used by: dvasin 38890 dvreasin 38892 dvreacos 38893

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