Theorem asindmre 33495

Description: Real part of domain of differentiability of arcsine. (Contributed by Brendan Leahy, 19-Dec-2018.)

Hypothesis

Ref	Expression
dvasin.d	|- D = ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )

Assertion

Ref	Expression
asindmre	|- ( D i^i RR ) = ( -u 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 3771	. . . . 5 |- ( ( -u 1 (,) 1 ) u. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( ( -oo (,] -u 1 ) u. ( ( -u 1 (,) 1 ) u. ( 1 [,) +oo ) ) )
2		neg1rr 11125	. . . . . . . . . 10 |- -u 1 e. RR
3	2	rexri 10097	. . . . . . . . 9 |- -u 1 e. RR*
4		1re 10039	. . . . . . . . . 10 |- 1 e. RR
5	4	rexri 10097	. . . . . . . . 9 |- 1 e. RR*
6		pnfxr 10092	. . . . . . . . 9 |- +oo e. RR*
7	3, 5, 6	3pm3.2i 1239	. . . . . . . 8 |- ( -u 1 e. RR* /\ 1 e. RR* /\ +oo e. RR* )
8		neg1lt0 11127	. . . . . . . . . 10 |- -u 1 < 0
9		0lt1 10550	. . . . . . . . . 10 |- 0 < 1
10		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
11	2, 10, 4	lttri 10163	. . . . . . . . . 10 |- ( ( -u 1 < 0 /\ 0 < 1 ) -> -u 1 < 1 )
12	8, 9, 11	mp2an 708	. . . . . . . . 9 |- -u 1 < 1
13		ltpnf 11954	. . . . . . . . . 10 |- ( 1 e. RR -> 1 < +oo )
14	4, 13	ax-mp 5	. . . . . . . . 9 |- 1 < +oo
15	12, 14	pm3.2i 471	. . . . . . . 8 |- ( -u 1 < 1 /\ 1 < +oo )
16		df-ioo 12179	. . . . . . . . 9 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
17		df-ico 12181	. . . . . . . . 9 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
18		xrlenlt 10103	. . . . . . . . 9 |- ( ( 1 e. RR* /\ w e. RR* ) -> ( 1 <_ w <-> -. w < 1 ) )
19		xrlttr 11973	. . . . . . . . 9 |- ( ( w e. RR* /\ 1 e. RR* /\ +oo e. RR* ) -> ( ( w < 1 /\ 1 < +oo ) -> w < +oo ) )
20		xrltletr 11988	. . . . . . . . 9 |- ( ( -u 1 e. RR* /\ 1 e. RR* /\ w e. RR* ) -> ( ( -u 1 < 1 /\ 1 <_ w ) -> -u 1 < w ) )
21	16, 17, 18, 16, 19, 20	ixxun 12191	. . . . . . . 8 |- ( ( ( -u 1 e. RR* /\ 1 e. RR* /\ +oo e. RR* ) /\ ( -u 1 < 1 /\ 1 < +oo ) ) -> ( ( -u 1 (,) 1 ) u. ( 1 [,) +oo ) ) = ( -u 1 (,) +oo ) )
22	7, 15, 21	mp2an 708	. . . . . . 7 |- ( ( -u 1 (,) 1 ) u. ( 1 [,) +oo ) ) = ( -u 1 (,) +oo )
23	22	uneq2i 3764	. . . . . 6 |- ( ( -oo (,] -u 1 ) u. ( ( -u 1 (,) 1 ) u. ( 1 [,) +oo ) ) ) = ( ( -oo (,] -u 1 ) u. ( -u 1 (,) +oo ) )
24		mnfxr 10096	. . . . . . . 8 |- -oo e. RR*
25	24, 3, 6	3pm3.2i 1239	. . . . . . 7 |- ( -oo e. RR* /\ -u 1 e. RR* /\ +oo e. RR* )
26		mnflt 11957	. . . . . . . . 9 |- ( -u 1 e. RR -> -oo < -u 1 )
27		ltpnf 11954	. . . . . . . . 9 |- ( -u 1 e. RR -> -u 1 < +oo )
28	26, 27	jca 554	. . . . . . . 8 |- ( -u 1 e. RR -> ( -oo < -u 1 /\ -u 1 < +oo ) )
29	2, 28	ax-mp 5	. . . . . . 7 |- ( -oo < -u 1 /\ -u 1 < +oo )
30		df-ioc 12180	. . . . . . . 8 |- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } )
31		xrltnle 10105	. . . . . . . 8 |- ( ( -u 1 e. RR* /\ w e. RR* ) -> ( -u 1 < w <-> -. w <_ -u 1 ) )
32		xrlelttr 11987	. . . . . . . 8 |- ( ( w e. RR* /\ -u 1 e. RR* /\ +oo e. RR* ) -> ( ( w <_ -u 1 /\ -u 1 < +oo ) -> w < +oo ) )
33		xrlttr 11973	. . . . . . . 8 |- ( ( -oo e. RR* /\ -u 1 e. RR* /\ w e. RR* ) -> ( ( -oo < -u 1 /\ -u 1 < w ) -> -oo < w ) )
34	30, 16, 31, 16, 32, 33	ixxun 12191	. . . . . . 7 |- ( ( ( -oo e. RR* /\ -u 1 e. RR* /\ +oo e. RR* ) /\ ( -oo < -u 1 /\ -u 1 < +oo ) ) -> ( ( -oo (,] -u 1 ) u. ( -u 1 (,) +oo ) ) = ( -oo (,) +oo ) )
35	25, 29, 34	mp2an 708	. . . . . 6 |- ( ( -oo (,] -u 1 ) u. ( -u 1 (,) +oo ) ) = ( -oo (,) +oo )
36	23, 35	eqtri 2644	. . . . 5 |- ( ( -oo (,] -u 1 ) u. ( ( -u 1 (,) 1 ) u. ( 1 [,) +oo ) ) ) = ( -oo (,) +oo )
37		ioomax 12248	. . . . 5 |- ( -oo (,) +oo ) = RR
38	1, 36, 37	3eqtri 2648	. . . 4 |- ( ( -u 1 (,) 1 ) u. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = RR
39	38	difeq1i 3724	. . 3 |- ( ( ( -u 1 (,) 1 ) u. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( RR \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
40		difun2 4048	. . 3 |- ( ( ( -u 1 (,) 1 ) u. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( ( -u 1 (,) 1 ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
41		ax-resscn 9993	. . . 4 |- RR C_ CC
42		difin2 3890	. . . 4 |- ( RR C_ CC -> ( RR \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) i^i RR ) )
43	41, 42	ax-mp 5	. . 3 |- ( RR \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) i^i RR )
44	39, 40, 43	3eqtr3ri 2653	. 2 |- ( ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) i^i RR ) = ( ( -u 1 (,) 1 ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
45		dvasin.d	. . 3 |- D = ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
46	45	ineq1i 3810	. 2 |- ( D i^i RR ) = ( ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) i^i RR )
47		incom 3805	. . . . 5 |- ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) = ( ( -oo (,] -u 1 ) i^i ( -u 1 (,) 1 ) )
48	30, 16, 31	ixxdisj 12190	. . . . . 6 |- ( ( -oo e. RR* /\ -u 1 e. RR* /\ 1 e. RR* ) -> ( ( -oo (,] -u 1 ) i^i ( -u 1 (,) 1 ) ) = (/) )
49	24, 3, 5, 48	mp3an 1424	. . . . 5 |- ( ( -oo (,] -u 1 ) i^i ( -u 1 (,) 1 ) ) = (/)
50	47, 49	eqtri 2644	. . . 4 |- ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) = (/)
51	16, 17, 18	ixxdisj 12190	. . . . 5 |- ( ( -u 1 e. RR* /\ 1 e. RR* /\ +oo e. RR* ) -> ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) = (/) )
52	3, 5, 6, 51	mp3an 1424	. . . 4 |- ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) = (/)
53	50, 52	pm3.2i 471	. . 3 |- ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) = (/) /\ ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) = (/) )
54		un00 4011	. . . 4 |- ( ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) = (/) /\ ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) = (/) ) <-> ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) u. ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) ) = (/) )
55		indi 3873	. . . . 5 |- ( ( -u 1 (,) 1 ) i^i ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) u. ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) )
56	55	eqeq1i 2627	. . . 4 |- ( ( ( -u 1 (,) 1 ) i^i ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = (/) <-> ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) u. ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) ) = (/) )
57		disj3 4021	. . . 4 |- ( ( ( -u 1 (,) 1 ) i^i ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) = (/) <-> ( -u 1 (,) 1 ) = ( ( -u 1 (,) 1 ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) )
58	54, 56, 57	3bitr2i 288	. . 3 |- ( ( ( ( -u 1 (,) 1 ) i^i ( -oo (,] -u 1 ) ) = (/) /\ ( ( -u 1 (,) 1 ) i^i ( 1 [,) +oo ) ) = (/) ) <-> ( -u 1 (,) 1 ) = ( ( -u 1 (,) 1 ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) )
59	53, 58	mpbi 220	. 2 |- ( -u 1 (,) 1 ) = ( ( -u 1 (,) 1 ) \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
60	44, 46, 59	3eqtr4i 2654	1 |- ( D i^i RR ) = ( -u 1 (,) 1 )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   (,)cioo 12175   (,]cioc 12176   [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ioo 12179  df-ioc 12180  df-ico 12181

This theorem is referenced by:  dvasin  33496  dvreasin  33498  dvreacos  33499