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Theorem atans 20242
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 5881 . . . 4  |-  ( y  =  A  ->  (
y ^ 2 )  =  ( A ^
2 ) )
21oveq2d 5890 . . 3  |-  ( y  =  A  ->  (
1  +  ( y ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
32eleq1d 2362 . 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 2938 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 1632    e. wcel 1696   {crab 2560    \ cdif 3162  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    -oocmnf 8881   2c2 9811   (,]cioc 10673   ^cexp 11120
This theorem is referenced by:  atans2  20243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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