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Theorem atandm 20669
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 3290 . . 3  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i } ) )
2 elprg 3791 . . . . . 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 2652 . . . . 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 6065 . . . 4  |-  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  x )
) )  -  ( log `  ( 1  +  ( _i  x.  x
) ) ) ) )  e.  _V
9 df-atan 20660 . . . 4  |- arctan  =  ( x  e.  ( CC 
\  { -u _i ,  _i } )  |->  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  (
1  +  ( _i  x.  x ) ) ) ) ) )
108, 9dmmpti 5533 . . 3  |-  dom arctan  =  ( CC  \  { -u _i ,  _i }
)
1110eleq2i 2468 . 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 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {cpr 3775   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   logclog 20405  arctancatan 20657
This theorem is referenced by:  atandm2  20670  atandm3  20671  atancj  20703  2efiatan  20711  tanatan  20712  dvatan  20728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-ov 6043  df-atan 20660
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