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Theorem atandm 22290
Description: Since the property is a little lengthy, we abbreviate  A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i as  A  e.  dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
atandm  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )

Proof of Theorem atandm
StepHypRef Expression
1 eldif 3357 . . 3  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i } ) )
2 elprg 3912 . . . . . 6  |-  ( A  e.  CC  ->  ( A  e.  { -u _i ,  _i }  <->  ( A  =  -u _i  \/  A  =  _i ) ) )
32notbid 294 . . . . 5  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  -.  ( A  =  -u _i  \/  A  =  _i ) ) )
4 neanior 2716 . . . . 5  |-  ( ( A  =/=  -u _i  /\  A  =/=  _i ) 
<->  -.  ( A  = 
-u _i  \/  A  =  _i ) )
53, 4syl6bbr 263 . . . 4  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
65pm5.32i 637 . . 3  |-  ( ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i }
)  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
71, 6bitri 249 . 2  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
8 ovex 6135 . . . 4  |-  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  x )
) )  -  ( log `  ( 1  +  ( _i  x.  x
) ) ) ) )  e.  _V
9 df-atan 22281 . . . 4  |- arctan  =  ( x  e.  ( CC 
\  { -u _i ,  _i } )  |->  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  (
1  +  ( _i  x.  x ) ) ) ) ) )
108, 9dmmpti 5559 . . 3  |-  dom arctan  =  ( CC  \  { -u _i ,  _i }
)
1110eleq2i 2507 . 2  |-  ( A  e.  dom arctan  <->  A  e.  ( CC  \  { -u _i ,  _i } ) )
12 3anass 969 . 2  |-  ( ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
137, 11, 123bitr4i 277 1  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620    \ cdif 3344   {cpr 3898   dom cdm 4859   ` cfv 5437  (class class class)co 6110   CCcc 9299   1c1 9302   _ici 9303    + caddc 9304    x. cmul 9306    - cmin 9614   -ucneg 9615    / cdiv 10012   2c2 10390   logclog 22025  arctancatan 22278
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-iota 5400  df-fun 5439  df-fn 5440  df-fv 5445  df-ov 6113  df-atan 22281
This theorem is referenced by:  atandm2  22291  atandm3  22292  atancj  22324  2efiatan  22332  tanatan  22333  dvatan  22349
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