Theorem fourierdlem29 40353

Description: Explicit function value for K applied to A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
fourierdlem29.1	|- K = ( s e. ( -u pi [,] pi ) |-> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )

Assertion

Ref	Expression
fourierdlem29	|- ( A e. ( -u pi [,] pi ) -> ( K ` A ) = if ( A = 0 , 1 , ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) ) )

Distinct variable group:   A, s

Allowed substitution hint:   K( s)

Proof of Theorem fourierdlem29

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( s = A -> ( s = 0 <-> A = 0 ) )
2		id 22	. . . 4 |- ( s = A -> s = A )
3		oveq1 6657	. . . . . 6 |- ( s = A -> ( s / 2 ) = ( A / 2 ) )
4	3	fveq2d 6195	. . . . 5 |- ( s = A -> ( sin ` ( s / 2 ) ) = ( sin ` ( A / 2 ) ) )
5	4	oveq2d 6666	. . . 4 |- ( s = A -> ( 2 x. ( sin ` ( s / 2 ) ) ) = ( 2 x. ( sin ` ( A / 2 ) ) ) )
6	2, 5	oveq12d 6668	. . 3 |- ( s = A -> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) )
7	1, 6	ifbieq2d 4111	. 2 |- ( s = A -> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = if ( A = 0 , 1 , ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) ) )
8		fourierdlem29.1	. 2 |- K = ( s e. ( -u pi [,] pi ) |-> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )
9		1ex 10035	. . 3 |- 1 e. _V
10		ovex 6678	. . 3 |- ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) e. _V
11	9, 10	ifex 4156	. 2 |- if ( A = 0 , 1 , ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) ) e. _V
12	7, 8, 11	fvmpt 6282	1 |- ( A e. ( -u pi [,] pi ) -> ( K ` A ) = if ( A = 0 , 1 , ( A / ( 2 x. ( sin ` ( A / 2 ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   -ucneg 10267   / cdiv 10684   2c2 11070   [,]cicc 12178   sincsin 14794   picpi 14797

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  ax-1cn 9994

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

This theorem is referenced by: (None)