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Theorem asindmre 31730
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.
StepHypRef Expression
1 un12 3630 . . . . 5  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )
2 neg1rr 10714 . . . . . . . . . 10  |-  -u 1  e.  RR
32rexri 9692 . . . . . . . . 9  |-  -u 1  e.  RR*
4 1re 9641 . . . . . . . . . 10  |-  1  e.  RR
54rexri 9692 . . . . . . . . 9  |-  1  e.  RR*
6 pnfxr 11412 . . . . . . . . 9  |- +oo  e.  RR*
73, 5, 63pm3.2i 1183 . . . . . . . 8  |-  ( -u
1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )
8 neg1lt0 10716 . . . . . . . . . 10  |-  -u 1  <  0
9 0lt1 10135 . . . . . . . . . 10  |-  0  <  1
10 0re 9642 . . . . . . . . . . 11  |-  0  e.  RR
112, 10, 4lttri 9759 . . . . . . . . . 10  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
128, 9, 11mp2an 676 . . . . . . . . 9  |-  -u 1  <  1
13 ltpnf 11422 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
144, 13ax-mp 5 . . . . . . . . 9  |-  1  < +oo
1512, 14pm3.2i 456 . . . . . . . 8  |-  ( -u
1  <  1  /\  1  < +oo )
16 df-ioo 11639 . . . . . . . . 9  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
17 df-ico 11641 . . . . . . . . 9  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
18 xrlenlt 9698 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\  w  e.  RR* )  ->  (
1  <_  w  <->  -.  w  <  1 ) )
19 xrlttr 11439 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  1  /\  1  < +oo )  ->  w  < +oo ) )
20 xrltletr 11454 . . . . . . . . 9  |-  ( (
-u 1  e.  RR*  /\  1  e.  RR*  /\  w  e.  RR* )  ->  (
( -u 1  <  1  /\  1  <_  w )  ->  -u 1  <  w
) )
2116, 17, 18, 16, 19, 20ixxun 11651 . . . . . . . 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 ) )
227, 15, 21mp2an 676 . . . . . . 7  |-  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) )  =  ( -u 1 (,) +oo )
2322uneq2i 3623 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)
24 mnfxr 11414 . . . . . . . 8  |- -oo  e.  RR*
2524, 3, 63pm3.2i 1183 . . . . . . 7  |-  ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )
26 mnflt 11425 . . . . . . . . 9  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
27 ltpnf 11422 . . . . . . . . 9  |-  ( -u
1  e.  RR  ->  -u
1  < +oo )
2826, 27jca 534 . . . . . . . 8  |-  ( -u
1  e.  RR  ->  ( -oo  <  -u 1  /\  -u 1  < +oo )
)
292, 28ax-mp 5 . . . . . . 7  |-  ( -oo  <  -u 1  /\  -u 1  < +oo )
30 df-ioc 11640 . . . . . . . 8  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
31 xrltnle 9700 . . . . . . . 8  |-  ( (
-u 1  e.  RR*  /\  w  e.  RR* )  ->  ( -u 1  < 
w  <->  -.  w  <_  -u
1 ) )
32 xrlelttr 11453 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  -u 1  /\  -u 1  < +oo )  ->  w  < +oo )
)
33 xrlttr 11439 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  -u 1  /\  -u 1  <  w
)  -> -oo  <  w
) )
3430, 16, 31, 16, 32, 33ixxun 11651 . . . . . . 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 ) )
3525, 29, 34mp2an 676 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)  =  ( -oo (,) +oo )
3623, 35eqtri 2458 . . . . 5  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( -oo (,) +oo )
37 ioomax 11709 . . . . 5  |-  ( -oo (,) +oo )  =  RR
381, 36, 373eqtri 2462 . . . 4  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  RR
3938difeq1i 3585 . . 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 3881 . . 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 9595 . . . 4  |-  RR  C_  CC
42 difin2 3741 . . . 4  |-  ( RR  C_  CC  ->  ( RR  \  ( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  =  ( ( CC  \  (
( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  i^i  RR ) )
4341, 42ax-mp 5 . . 3  |-  ( RR 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )  =  ( ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  i^i  RR )
4439, 40, 433eqtr3ri 2467 . 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 ) ) )
4645ineq1i 3666 . 2  |-  ( D  i^i  RR )  =  ( ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  i^i  RR )
47 incom 3661 . . . . 5  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  ( ( -oo (,] -u 1 )  i^i  ( -u 1 (,) 1 ) )
4830, 16, 31ixxdisj 11650 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\  1  e. 
RR* )  ->  (
( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/) )
4924, 3, 5, 48mp3an 1360 . . . . 5  |-  ( ( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/)
5047, 49eqtri 2458 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  (/)
5116, 17, 18ixxdisj 11650 . . . . 5  |-  ( (
-u 1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  (
( -u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/) )
523, 5, 6, 51mp3an 1360 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/)
5350, 52pm3.2i 456 . . 3  |-  ( ( ( -u 1 (,) 1 )  i^i  ( -oo (,] -u 1 ) )  =  (/)  /\  (
( -u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/) )
54 un00 3834 . . . 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 3725 . . . . 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 )
) )
5655eqeq1i 2436 . . . 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 3843 . . . 4  |-  ( ( ( -u 1 (,) 1 )  i^i  (
( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  (/)  <->  ( -u 1 (,) 1 )  =  ( ( -u 1 (,) 1 )  \  (
( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) ) )
5854, 56, 573bitr2i 276 . . 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 ) ) ) )
5953, 58mpbi 211 . 2  |-  ( -u
1 (,) 1 )  =  ( ( -u
1 (,) 1 ) 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
6044, 46, 593eqtr4i 2468 1  |-  ( D  i^i  RR )  =  ( -u 1 (,) 1 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   class class class wbr 4426  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673    < clt 9674    <_ cle 9675   -ucneg 9860   (,)cioo 11635   (,]cioc 11636   [,)cico 11637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-ioo 11639  df-ioc 11640  df-ico 11641
This theorem is referenced by:  dvasin  31731  dvreasin  31733  dvreacos  31734
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