Theorem acosval 24610

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( x = A -> (arcsin ` x ) = (arcsin ` A ) )
2	1	oveq2d 6666	. 2 |- ( x = A -> ( ( pi / 2 ) - (arcsin ` x ) ) = ( ( pi / 2 ) - (arcsin ` A ) ) )
3		df-acos 24593	. 2 |- arccos = ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) )
4		ovex 6678	. 2 |- ( ( pi / 2 ) - (arcsin ` A ) ) e. _V
5	2, 3, 4	fvmpt 6282	1 |- ( A e. CC -> (arccos ` A ) = ( ( pi / 2 ) - (arcsin ` A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   - cmin 10266   / cdiv 10684   2c2 11070   picpi 14797  arcsincasin 24589  arccoscacos 24590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-acos 24593

This theorem is referenced by:  acosneg  24614  cosacos  24617  acoscos  24620  acos1  24622  acosbnd  24627  acosrecl  24630  sinacos  24632