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Theorem atandm 23323
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 3399 . . 3  |-  ( A  e.  ( CC  \  { -u _i ,  _i } )  <->  ( A  e.  CC  /\  -.  A  e.  { -u _i ,  _i } ) )
2 elprg 3960 . . . . . 6  |-  ( A  e.  CC  ->  ( A  e.  { -u _i ,  _i }  <->  ( A  =  -u _i  \/  A  =  _i ) ) )
32notbid 292 . . . . 5  |-  ( A  e.  CC  ->  ( -.  A  e.  { -u _i ,  _i }  <->  -.  ( A  =  -u _i  \/  A  =  _i ) ) )
4 neanior 2707 . . . . 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 635 . . 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 6224 . . . 4  |-  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  x )
) )  -  ( log `  ( 1  +  ( _i  x.  x
) ) ) ) )  e.  _V
9 df-atan 23314 . . . 4  |- arctan  =  ( x  e.  ( CC 
\  { -u _i ,  _i } )  |->  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  x ) ) )  -  ( log `  (
1  +  ( _i  x.  x ) ) ) ) ) )
108, 9dmmpti 5618 . . 3  |-  dom arctan  =  ( CC  \  { -u _i ,  _i }
)
1110eleq2i 2460 . 2  |-  ( A  e.  dom arctan  <->  A  e.  ( CC  \  { -u _i ,  _i } ) )
12 3anass 975 . 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386   {cpr 3946   dom cdm 4913   ` cfv 5496  (class class class)co 6196   CCcc 9401   1c1 9404   _ici 9405    + caddc 9406    x. cmul 9408    - cmin 9718   -ucneg 9719    / cdiv 10123   2c2 10502   logclog 23027  arctancatan 23311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-ov 6199  df-atan 23314
This theorem is referenced by:  atandm2  23324  atandm3  23325  atancj  23357  2efiatan  23365  tanatan  23366  dvatan  23382
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