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Theorem atandm 20716
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 3330 . . 3  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i } ) )
2 elprg 3831 . . . . . 6  |-  ( A  e.  CC  ->  ( A  e.  { -u _i ,  _i }  <->  ( A  =  -u _i  \/  A  =  _i ) ) )
32notbid 286 . . . . 5  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  -.  ( A  =  -u _i  \/  A  =  _i ) ) )
4 neanior 2689 . . . . 5  |-  ( ( A  =/=  -u _i  /\  A  =/=  _i ) 
<->  -.  ( A  = 
-u _i  \/  A  =  _i ) )
53, 4syl6bbr 255 . . . 4  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
65pm5.32i 619 . . 3  |-  ( ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i }
)  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
71, 6bitri 241 . 2  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
8 ovex 6106 . . . 4  |-  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  x )
) )  -  ( log `  ( 1  +  ( _i  x.  x
) ) ) ) )  e.  _V
9 df-atan 20707 . . . 4  |- arctan  =  ( x  e.  ( CC 
\  { -u _i ,  _i } )  |->  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  (
1  +  ( _i  x.  x ) ) ) ) ) )
108, 9dmmpti 5574 . . 3  |-  dom arctan  =  ( CC  \  { -u _i ,  _i }
)
1110eleq2i 2500 . 2  |-  ( A  e.  dom arctan  <->  A  e.  ( CC  \  { -u _i ,  _i } ) )
12 3anass 940 . 2  |-  ( ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
137, 11, 123bitr4i 269 1  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {cpr 3815   dom cdm 4878   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   logclog 20452  arctancatan 20704
This theorem is referenced by:  atandm2  20717  atandm3  20718  atancj  20750  2efiatan  20758  tanatan  20759  dvatan  20775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-ov 6084  df-atan 20707
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