Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  asindmre Structured version   Unicode version

Theorem asindmre 29695
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 3662 . . . . 5  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )
2 neg1rr 10639 . . . . . . . . . 10  |-  -u 1  e.  RR
32rexri 9645 . . . . . . . . 9  |-  -u 1  e.  RR*
4 1re 9594 . . . . . . . . . 10  |-  1  e.  RR
54rexri 9645 . . . . . . . . 9  |-  1  e.  RR*
6 pnfxr 11320 . . . . . . . . 9  |- +oo  e.  RR*
73, 5, 63pm3.2i 1174 . . . . . . . 8  |-  ( -u
1  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )
8 neg1lt0 10641 . . . . . . . . . 10  |-  -u 1  <  0
9 0lt1 10074 . . . . . . . . . 10  |-  0  <  1
10 0re 9595 . . . . . . . . . . 11  |-  0  e.  RR
112, 10, 4lttri 9709 . . . . . . . . . 10  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
128, 9, 11mp2an 672 . . . . . . . . 9  |-  -u 1  <  1
13 ltpnf 11330 . . . . . . . . . 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 11532 . . . . . . . . 9  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
17 df-ico 11534 . . . . . . . . 9  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
18 xrlenlt 9651 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\  w  e.  RR* )  ->  (
1  <_  w  <->  -.  w  <  1 ) )
19 xrlttr 11345 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  1  /\  1  < +oo )  ->  w  < +oo ) )
20 xrltletr 11359 . . . . . . . . 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 11544 . . . . . . . 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 3655 . . . . . 6  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( ( -oo (,] -u 1
)  u.  ( -u
1 (,) +oo )
)
24 mnfxr 11322 . . . . . . . 8  |- -oo  e.  RR*
2524, 3, 63pm3.2i 1174 . . . . . . 7  |-  ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )
26 mnflt 11332 . . . . . . . . 9  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
27 ltpnf 11330 . . . . . . . . 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 11533 . . . . . . . 8  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
31 xrltnle 9652 . . . . . . . 8  |-  ( (
-u 1  e.  RR*  /\  w  e.  RR* )  ->  ( -u 1  < 
w  <->  -.  w  <_  -u
1 ) )
32 xrlelttr 11358 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  -u 1  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  -u 1  /\  -u 1  < +oo )  ->  w  < +oo )
)
33 xrlttr 11345 . . . . . . . 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 11544 . . . . . . 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 2496 . . . . 5  |-  ( ( -oo (,] -u 1
)  u.  ( (
-u 1 (,) 1
)  u.  ( 1 [,) +oo ) ) )  =  ( -oo (,) +oo )
37 ioomax 11598 . . . . 5  |-  ( -oo (,) +oo )  =  RR
381, 36, 373eqtri 2500 . . . 4  |-  ( (
-u 1 (,) 1
)  u.  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )  =  RR
3938difeq1i 3618 . . 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 3906 . . 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 9548 . . . 4  |-  RR  C_  CC
42 difin2 3760 . . . 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 2505 . 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 3696 . 2  |-  ( D  i^i  RR )  =  ( ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  i^i  RR )
47 incom 3691 . . . . 5  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  ( ( -oo (,] -u 1 )  i^i  ( -u 1 (,) 1 ) )
4830, 16, 31ixxdisj 11543 . . . . . 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 2496 . . . 4  |-  ( (
-u 1 (,) 1
)  i^i  ( -oo (,] -u 1 ) )  =  (/)
5116, 17, 18ixxdisj 11543 . . . . 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 3862 . . . 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 3744 . . . . 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 2474 . . . 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 3871 . . . 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 2506 1  |-  ( D  i^i  RR )  =  ( -u 1 (,) 1 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626    < clt 9627    <_ cle 9628   -ucneg 9805   (,)cioo 11528   (,]cioc 11529   [,)cico 11530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-ioo 11532  df-ioc 11533  df-ico 11534
This theorem is referenced by:  dvasin  29696  dvreasin  29698  dvreacos  29699
  Copyright terms: Public domain W3C validator