asindmre - Mathbox for Brendan Leahy

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Theorem asindmre 37954

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 4127	. . . . 5 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞)))
2		neg1rr 12143	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
3	2	rexri 11202	. . . . . . . . 9 ⊢ -1 ∈ ℝ^*
4		1xr 11203	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
5		pnfxr 11198	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
6	3, 4, 5	3pm3.2i 1341	. . . . . . . 8 ⊢ (-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
7		neg1lt0 12145	. . . . . . . . . 10 ⊢ -1 < 0
8		0lt1 11671	. . . . . . . . . 10 ⊢ 0 < 1
9		0re 11146	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
10		1re 11144	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
11	2, 9, 10	lttri 11271	. . . . . . . . . 10 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
12	7, 8, 11	mp2an 693	. . . . . . . . 9 ⊢ -1 < 1
13		ltpnf 13046	. . . . . . . . . 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 13277	. . . . . . . . 9 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
17		df-ico 13279	. . . . . . . . 9 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18		xrlenlt 11209	. . . . . . . . 9 ⊢ ((1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (1 ≤ 𝑤 ↔ ¬ 𝑤 < 1))
19		xrlttr 13066	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 1 ∧ 1 < +∞) → 𝑤 < +∞))
20		xrltletr 13083	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-1 < 1 ∧ 1 ≤ 𝑤) → -1 < 𝑤))
21	16, 17, 18, 16, 19, 20	ixxun 13289	. . . . . . . 8 ⊢ (((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-1 < 1 ∧ 1 < +∞)) → ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞))
22	6, 15, 21	mp2an 693	. . . . . . 7 ⊢ ((-1(,)1) ∪ (1[,)+∞)) = (-1(,)+∞)
23	22	uneq2i 4119	. . . . . 6 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = ((-∞(,]-1) ∪ (-1(,)+∞))
24		mnfxr 11201	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
25	24, 3, 5	3pm3.2i 1341	. . . . . . 7 ⊢ (-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
26		mnflt 13049	. . . . . . . . 9 ⊢ (-1 ∈ ℝ → -∞ < -1)
27		ltpnf 13046	. . . . . . . . 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 13278	. . . . . . . 8 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
31		xrltnle 11211	. . . . . . . 8 ⊢ ((-1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (-1 < 𝑤 ↔ ¬ 𝑤 ≤ -1))
32		xrlelttr 13082	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 ≤ -1 ∧ -1 < +∞) → 𝑤 < +∞))
33		xrlttr 13066	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < -1 ∧ -1 < 𝑤) → -∞ < 𝑤))
34	30, 16, 31, 16, 32, 33	ixxun 13289	. . . . . . 7 ⊢ (((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < -1 ∧ -1 < +∞)) → ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞))
35	25, 29, 34	mp2an 693	. . . . . 6 ⊢ ((-∞(,]-1) ∪ (-1(,)+∞)) = (-∞(,)+∞)
36	23, 35	eqtri 2760	. . . . 5 ⊢ ((-∞(,]-1) ∪ ((-1(,)1) ∪ (1[,)+∞))) = (-∞(,)+∞)
37		ioomax 13350	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
38	1, 36, 37	3eqtri 2764	. . . 4 ⊢ ((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) = ℝ
39	38	difeq1i 4076	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = (ℝ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
40		difun2 4435	. . 3 ⊢ (((-1(,)1) ∪ ((-∞(,]-1) ∪ (1[,)+∞))) ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
41		ax-resscn 11095	. . . 4 ⊢ ℝ ⊆ ℂ
42		difin2 4255	. . . 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 2769	. 2 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = ((-1(,)1) ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
45		dvasin.d	. . 3 ⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))
46	45	ineq1i 4170	. 2 ⊢ (𝐷 ∩ ℝ) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ)
47		incom 4163	. . . . 5 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ((-∞(,]-1) ∩ (-1(,)1))
48	30, 16, 31	ixxdisj 13288	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ -1 ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((-∞(,]-1) ∩ (-1(,)1)) = ∅)
49	24, 3, 4, 48	mp3an 1464	. . . . 5 ⊢ ((-∞(,]-1) ∩ (-1(,)1)) = ∅
50	47, 49	eqtri 2760	. . . 4 ⊢ ((-1(,)1) ∩ (-∞(,]-1)) = ∅
51	16, 17, 18	ixxdisj 13288	. . . . 5 ⊢ ((-1 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-1(,)1) ∩ (1[,)+∞)) = ∅)
52	3, 4, 5, 51	mp3an 1464	. . . 4 ⊢ ((-1(,)1) ∩ (1[,)+∞)) = ∅
53	50, 52	pm3.2i 470	. . 3 ⊢ (((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅)
54		un00 4399	. . . 4 ⊢ ((((-1(,)1) ∩ (-∞(,]-1)) = ∅ ∧ ((-1(,)1) ∩ (1[,)+∞)) = ∅) ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
55		indi 4238	. . . . 5 ⊢ ((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞)))
56	55	eqeq1i 2742	. . . 4 ⊢ (((-1(,)1) ∩ ((-∞(,]-1) ∪ (1[,)+∞))) = ∅ ↔ (((-1(,)1) ∩ (-∞(,]-1)) ∪ ((-1(,)1) ∩ (1[,)+∞))) = ∅)
57		disj3 4408	. . . 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 2770	1 ⊢ (𝐷 ∩ ℝ) = (-1(,)1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 +∞cpnf 11175 -∞cmnf 11176 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 -cneg 11377 (,)cioo 13273 (,]cioc 13274 [,)cico 13275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-ioo 13277 df-ioc 13278 df-ico 13279

This theorem is referenced by: dvasin 37955 dvreasin 37957 dvreacos 37958

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