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Theorem asindmre 30286
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 3658 . . . . 5  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )
2 neg1rr 10661 . . . . . . . . . 10  |-  -u 1  e.  RR
32rexri 9663 . . . . . . . . 9  |-  -u 1  e.  RR*
4 1re 9612 . . . . . . . . . 10  |-  1  e.  RR
54rexri 9663 . . . . . . . . 9  |-  1  e.  RR*
6 pnfxr 11346 . . . . . . . . 9  |- +oo  e.  RR*
73, 5, 63pm3.2i 1174 . . . . . . . 8  |-  ( -u
1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )
8 neg1lt0 10663 . . . . . . . . . 10  |-  -u 1  <  0
9 0lt1 10096 . . . . . . . . . 10  |-  0  <  1
10 0re 9613 . . . . . . . . . . 11  |-  0  e.  RR
112, 10, 4lttri 9727 . . . . . . . . . 10  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
128, 9, 11mp2an 672 . . . . . . . . 9  |-  -u 1  <  1
13 ltpnf 11356 . . . . . . . . . 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 11558 . . . . . . . . 9  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
17 df-ico 11560 . . . . . . . . 9  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
18 xrlenlt 9669 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\  w  e.  RR* )  ->  (
1  <_  w  <->  -.  w  <  1 ) )
19 xrlttr 11371 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  1  /\  1  < +oo )  ->  w  < +oo ) )
20 xrltletr 11385 . . . . . . . . 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 11570 . . . . . . . 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 3651 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)
24 mnfxr 11348 . . . . . . . 8  |- -oo  e.  RR*
2524, 3, 63pm3.2i 1174 . . . . . . 7  |-  ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )
26 mnflt 11358 . . . . . . . . 9  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
27 ltpnf 11356 . . . . . . . . 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 11559 . . . . . . . 8  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
31 xrltnle 9670 . . . . . . . 8  |-  ( (
-u 1  e.  RR*  /\  w  e.  RR* )  ->  ( -u 1  < 
w  <->  -.  w  <_  -u
1 ) )
32 xrlelttr 11384 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  -u 1  /\  -u 1  < +oo )  ->  w  < +oo )
)
33 xrlttr 11371 . . . . . . . 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 11570 . . . . . . 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 2486 . . . . 5  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( -oo (,) +oo )
37 ioomax 11624 . . . . 5  |-  ( -oo (,) +oo )  =  RR
381, 36, 373eqtri 2490 . . . 4  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  RR
3938difeq1i 3614 . . 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 3910 . . 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 9566 . . . 4  |-  RR  C_  CC
42 difin2 3767 . . . 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 2495 . 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 3692 . 2  |-  ( D  i^i  RR )  =  ( ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  i^i  RR )
47 incom 3687 . . . . 5  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  ( ( -oo (,] -u 1 )  i^i  ( -u 1 (,) 1 ) )
4830, 16, 31ixxdisj 11569 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\  1  e. 
RR* )  ->  (
( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/) )
4924, 3, 5, 48mp3an 1324 . . . . 5  |-  ( ( -oo (,] -u 1
)  i^i  ( -u 1 (,) 1 ) )  =  (/)
5047, 49eqtri 2486 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  (/)
5116, 17, 18ixxdisj 11569 . . . . 5  |-  ( (
-u 1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  (
( -u 1 (,) 1
)  i^i  ( 1 [,) +oo ) )  =  (/) )
523, 5, 6, 51mp3an 1324 . . . 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 3865 . . . 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 3751 . . . . 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 2464 . . . 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 3874 . . . 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 2496 1  |-  ( D  i^i  RR )  =  ( -u 1 (,) 1 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646   -ucneg 9825   (,)cioo 11554   (,]cioc 11555   [,)cico 11556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ioo 11558  df-ioc 11559  df-ico 11560
This theorem is referenced by:  dvasin  30287  dvreasin  30289  dvreacos  30290
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