Theorem dirkerval 40308

Description: The N_th Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
dirkerval.1	|- D = ( n e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )

Assertion

Ref	Expression
dirkerval	|- ( N e. NN -> ( D ` N ) = ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )

Distinct variable groups:   N, s   n, s

Allowed substitution hints:   D( n, s)   N( n)

Proof of Theorem dirkerval

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( m = N /\ s e. RR ) -> m = N )
2	1	oveq2d 6666	. . . . . 6 |- ( ( m = N /\ s e. RR ) -> ( 2 x. m ) = ( 2 x. N ) )
3	2	oveq1d 6665	. . . . 5 |- ( ( m = N /\ s e. RR ) -> ( ( 2 x. m ) + 1 ) = ( ( 2 x. N ) + 1 ) )
4	3	oveq1d 6665	. . . 4 |- ( ( m = N /\ s e. RR ) -> ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) = ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) )
5	1	oveq1d 6665	. . . . . . 7 |- ( ( m = N /\ s e. RR ) -> ( m + ( 1 / 2 ) ) = ( N + ( 1 / 2 ) ) )
6	5	oveq1d 6665	. . . . . 6 |- ( ( m = N /\ s e. RR ) -> ( ( m + ( 1 / 2 ) ) x. s ) = ( ( N + ( 1 / 2 ) ) x. s ) )
7	6	fveq2d 6195	. . . . 5 |- ( ( m = N /\ s e. RR ) -> ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) = ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) )
8	7	oveq1d 6665	. . . 4 |- ( ( m = N /\ s e. RR ) -> ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) )
9	4, 8	ifeq12d 4106	. . 3 |- ( ( m = N /\ s e. RR ) -> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) )
10	9	mpteq2dva 4744	. 2 |- ( m = N -> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) = ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )
11		dirkerval.1	. . 3 |- D = ( n e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )
12		simpl 473	. . . . . . . . 9 |- ( ( n = m /\ s e. RR ) -> n = m )
13	12	oveq2d 6666	. . . . . . . 8 |- ( ( n = m /\ s e. RR ) -> ( 2 x. n ) = ( 2 x. m ) )
14	13	oveq1d 6665	. . . . . . 7 |- ( ( n = m /\ s e. RR ) -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. m ) + 1 ) )
15	14	oveq1d 6665	. . . . . 6 |- ( ( n = m /\ s e. RR ) -> ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) = ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) )
16	12	oveq1d 6665	. . . . . . . . 9 |- ( ( n = m /\ s e. RR ) -> ( n + ( 1 / 2 ) ) = ( m + ( 1 / 2 ) ) )
17	16	oveq1d 6665	. . . . . . . 8 |- ( ( n = m /\ s e. RR ) -> ( ( n + ( 1 / 2 ) ) x. s ) = ( ( m + ( 1 / 2 ) ) x. s ) )
18	17	fveq2d 6195	. . . . . . 7 |- ( ( n = m /\ s e. RR ) -> ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) = ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) )
19	18	oveq1d 6665	. . . . . 6 |- ( ( n = m /\ s e. RR ) -> ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) = ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) )
20	15, 19	ifeq12d 4106	. . . . 5 |- ( ( n = m /\ s e. RR ) -> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) = if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) )
21	20	mpteq2dva 4744	. . . 4 |- ( n = m -> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) = ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )
22	21	cbvmptv 4750	. . 3 |- ( n e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) ) = ( m e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )
23	11, 22	eqtri 2644	. 2 |- D = ( m e. NN |-> ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. m ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( m + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )
24		reex 10027	. . 3 |- RR e. _V
25	24	mptex 6486	. 2 |- ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) e. _V
26	10, 23, 25	fvmpt 6282	1 |- ( N e. NN -> ( D ` N ) = ( s e. RR |-> if ( ( s mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) / ( ( 2 x. pi ) x. ( sin ` ( s / 2 ) ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   NNcn 11020   2c2 11070   mod cmo 12668   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653

This theorem is referenced by:  dirkerval2  40311  dirkerf  40314  dirkertrigeq  40318  dirkercncflem2  40321  dirkercncflem4  40323