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Theorem acosval 20179
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 5525 . . 3  |-  ( x  =  A  ->  (arcsin `  x )  =  (arcsin `  A ) )
21oveq2d 5874 . 2  |-  ( x  =  A  ->  (
( pi  /  2
)  -  (arcsin `  x ) )  =  ( ( pi  / 
2 )  -  (arcsin `  A ) ) )
3 df-acos 20162 . 2  |- arccos  =  ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )
4 ovex 5883 . 2  |-  ( ( pi  /  2 )  -  (arcsin `  A
) )  e.  _V
52, 3, 4fvmpt 5602 1  |-  ( A  e.  CC  ->  (arccos `  A )  =  ( ( pi  /  2
)  -  (arcsin `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735    - cmin 9037    / cdiv 9423   2c2 9795   picpi 12348  arcsincasin 20158  arccoscacos 20159
This theorem is referenced by:  acosneg  20183  cosacos  20186  acoscos  20189  acos1  20191  acosbnd  20196  acosrecl  20199  sinacos  20201  dvreacos  24924
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-acos 20162
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