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Theorem atans 20226
Description: The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
Hypotheses
Ref Expression
atansopn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
atansopn.s  |-  S  =  { y  e.  CC  |  ( 1  +  ( y ^ 2 ) )  e.  D }
Assertion
Ref Expression
atans  |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  e.  D ) )
Distinct variable groups:    y, A    y, D
Allowed substitution hint:    S( y)

Proof of Theorem atans
StepHypRef Expression
1 oveq1 5865 . . . 4  |-  ( y  =  A  ->  (
y ^ 2 )  =  ( A ^
2 ) )
21oveq2d 5874 . . 3  |-  ( y  =  A  ->  (
1  +  ( y ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
32eleq1d 2349 . 2  |-  ( y  =  A  ->  (
( 1  +  ( y ^ 2 ) )  e.  D  <->  ( 1  +  ( A ^
2 ) )  e.  D ) )
4 atansopn.s . 2  |-  S  =  { y  e.  CC  |  ( 1  +  ( y ^ 2 ) )  e.  D }
53, 4elrab2 2925 1  |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    -oocmnf 8865   2c2 9795   (,]cioc 10657   ^cexp 11104
This theorem is referenced by:  atans2  20227
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-ov 5861
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