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Theorem atandm 20172
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 3162 . . 3  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i } ) )
2 elprg 3657 . . . . . 6  |-  ( A  e.  CC  ->  ( A  e.  { -u _i ,  _i }  <->  ( A  =  -u _i  \/  A  =  _i ) ) )
32notbid 285 . . . . 5  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  -.  ( A  =  -u _i  \/  A  =  _i ) ) )
4 neanior 2531 . . . . 5  |-  ( ( A  =/=  -u _i  /\  A  =/=  _i ) 
<->  -.  ( A  = 
-u _i  \/  A  =  _i ) )
53, 4syl6bbr 254 . . . 4  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
65pm5.32i 618 . . 3  |-  ( ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i }
)  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
71, 6bitri 240 . 2  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
8 ovex 5883 . . . 4  |-  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  x )
) )  -  ( log `  ( 1  +  ( _i  x.  x
) ) ) ) )  e.  _V
9 df-atan 20163 . . . 4  |- arctan  =  ( x  e.  ( CC 
\  { -u _i ,  _i } )  |->  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  (
1  +  ( _i  x.  x ) ) ) ) ) )
108, 9dmmpti 5373 . . 3  |-  dom arctan  =  ( CC  \  { -u _i ,  _i }
)
1110eleq2i 2347 . 2  |-  ( A  e.  dom arctan  <->  A  e.  ( CC  \  { -u _i ,  _i } ) )
12 3anass 938 . 2  |-  ( ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
137, 11, 123bitr4i 268 1  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {cpr 3641   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   logclog 19912  arctancatan 20160
This theorem is referenced by:  atandm2  20173  atandm3  20174  atancj  20206  2efiatan  20214  tanatan  20215  dvatan  20231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-atan 20163
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