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Theorem atans 20772
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 6090 . . . 4  |-  ( y  =  A  ->  (
y ^ 2 )  =  ( A ^
2 ) )
21oveq2d 6099 . . 3  |-  ( y  =  A  ->  (
1  +  ( y ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
32eleq1d 2504 . 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 3096 1  |-  ( A  e.  S  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    \ cdif 3319  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    -oocmnf 9120   2c2 10051   (,]cioc 10919   ^cexp 11384
This theorem is referenced by:  atans2  20773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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