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Theorem acosval 20195
Description: Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
acosval  |-  ( A  e.  CC  ->  (arccos `  A )  =  ( ( pi  /  2
)  -  (arcsin `  A ) ) )

Proof of Theorem acosval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . 3  |-  ( x  =  A  ->  (arcsin `  x )  =  (arcsin `  A ) )
21oveq2d 5890 . 2  |-  ( x  =  A  ->  (
( pi  /  2
)  -  (arcsin `  x ) )  =  ( ( pi  / 
2 )  -  (arcsin `  A ) ) )
3 df-acos 20178 . 2  |- arccos  =  ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )
4 ovex 5899 . 2  |-  ( ( pi  /  2 )  -  (arcsin `  A
) )  e.  _V
52, 3, 4fvmpt 5618 1  |-  ( A  e.  CC  ->  (arccos `  A )  =  ( ( pi  /  2
)  -  (arcsin `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751    - cmin 9053    / cdiv 9439   2c2 9811   picpi 12364  arcsincasin 20174  arccoscacos 20175
This theorem is referenced by:  acosneg  20199  cosacos  20202  acoscos  20205  acos1  20207  acosbnd  20212  acosrecl  20215  sinacos  20217  dvreacos  25027
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-acos 20178
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