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Theorem asindmre 28479
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 3514 . . . . 5  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )
2 neg1rr 10426 . . . . . . . . . 10  |-  -u 1  e.  RR
32rexri 9436 . . . . . . . . 9  |-  -u 1  e.  RR*
4 1re 9385 . . . . . . . . . 10  |-  1  e.  RR
54rexri 9436 . . . . . . . . 9  |-  1  e.  RR*
6 pnfxr 11092 . . . . . . . . 9  |- +oo  e.  RR*
73, 5, 63pm3.2i 1166 . . . . . . . 8  |-  ( -u
1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )
8 neg1lt0 10428 . . . . . . . . . 10  |-  -u 1  <  0
9 0lt1 9862 . . . . . . . . . 10  |-  0  <  1
10 0re 9386 . . . . . . . . . . 11  |-  0  e.  RR
112, 10, 4lttri 9500 . . . . . . . . . 10  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
128, 9, 11mp2an 672 . . . . . . . . 9  |-  -u 1  <  1
13 ltpnf 11102 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
144, 13ax-mp 5 . . . . . . . . 9  |-  1  < +oo
1512, 14pm3.2i 455 . . . . . . . 8  |-  ( -u
1  <  1  /\  1  < +oo )
16 df-ioo 11304 . . . . . . . . 9  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
17 df-ico 11306 . . . . . . . . 9  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
18 xrlenlt 9442 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\  w  e.  RR* )  ->  (
1  <_  w  <->  -.  w  <  1 ) )
19 xrlttr 11117 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  1  /\  1  < +oo )  ->  w  < +oo ) )
20 xrltletr 11131 . . . . . . . . 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 11316 . . . . . . . 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 672 . . . . . . 7  |-  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) )  =  ( -u 1 (,) +oo )
2322uneq2i 3507 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)
24 mnfxr 11094 . . . . . . . 8  |- -oo  e.  RR*
2524, 3, 63pm3.2i 1166 . . . . . . 7  |-  ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )
26 mnflt 11104 . . . . . . . . 9  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
27 ltpnf 11102 . . . . . . . . 9  |-  ( -u
1  e.  RR  ->  -u
1  < +oo )
2826, 27jca 532 . . . . . . . 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 11305 . . . . . . . 8  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
31 xrltnle 9443 . . . . . . . 8  |-  ( (
-u 1  e.  RR*  /\  w  e.  RR* )  ->  ( -u 1  < 
w  <->  -.  w  <_  -u
1 ) )
32 xrlelttr 11130 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  -u 1  /\  -u 1  < +oo )  ->  w  < +oo )
)
33 xrlttr 11117 . . . . . . . 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 11316 . . . . . . 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 672 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)  =  ( -oo (,) +oo )
3623, 35eqtri 2463 . . . . 5  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( -oo (,) +oo )
37 ioomax 11370 . . . . 5  |-  ( -oo (,) +oo )  =  RR
381, 36, 373eqtri 2467 . . . 4  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  RR
3938difeq1i 3470 . . 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 3758 . . 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 9339 . . . 4  |-  RR  C_  CC
42 difin2 3612 . . . 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 2472 . 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 3548 . 2  |-  ( D  i^i  RR )  =  ( ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  i^i  RR )
47 incom 3543 . . . . 5  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  ( ( -oo (,] -u 1 )  i^i  ( -u 1 (,) 1 ) )
4830, 16, 31ixxdisj 11315 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\  1  e. 
RR* )  ->  (
( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/) )
4924, 3, 5, 48mp3an 1314 . . . . 5  |-  ( ( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/)
5047, 49eqtri 2463 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  (/)
5116, 17, 18ixxdisj 11315 . . . . 5  |-  ( (
-u 1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  (
( -u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/) )
523, 5, 6, 51mp3an 1314 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/)
5350, 52pm3.2i 455 . . 3  |-  ( ( ( -u 1 (,) 1 )  i^i  ( -oo (,] -u 1 ) )  =  (/)  /\  (
( -u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/) )
54 un00 3714 . . . 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 3596 . . . . 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 2450 . . . 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 3723 . . . 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 273 . . 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 208 . 2  |-  ( -u
1 (,) 1 )  =  ( ( -u
1 (,) 1 ) 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
6044, 46, 593eqtr4i 2473 1  |-  ( D  i^i  RR )  =  ( -u 1 (,) 1 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3325    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   class class class wbr 4292  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   -ucneg 9596   (,)cioo 11300   (,]cioc 11301   [,)cico 11302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-ioo 11304  df-ioc 11305  df-ico 11306
This theorem is referenced by:  dvasin  28480  dvreasin  28482  dvreacos  28483
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