Theorem fourierdlem3 40327

Description: Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
fourierdlem3.1	|- P = ( m e. NN |-> { p e. ( ( -u pi [,] pi ) ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` m ) = pi ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )

Assertion

Ref	Expression
fourierdlem3	|- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )

Distinct variable groups:   i, M, m, p   Q, i, p

Allowed substitution hints:   P( i, m, p)   Q( m)

Proof of Theorem fourierdlem3

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( m = M -> ( 0 ... m ) = ( 0 ... M ) )
2	1	oveq2d 6666	. . . . 5 |- ( m = M -> ( ( -u pi [,] pi ) ^m ( 0 ... m ) ) = ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) )
3		fveq2 6191	. . . . . . . 8 |- ( m = M -> ( p ` m ) = ( p ` M ) )
4	3	eqeq1d 2624	. . . . . . 7 |- ( m = M -> ( ( p ` m ) = pi <-> ( p ` M ) = pi ) )
5	4	anbi2d 740	. . . . . 6 |- ( m = M -> ( ( ( p ` 0 ) = -u pi /\ ( p ` m ) = pi ) <-> ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) ) )
6		oveq2 6658	. . . . . . 7 |- ( m = M -> ( 0..^ m ) = ( 0..^ M ) )
7	6	raleqdv 3144	. . . . . 6 |- ( m = M -> ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) )
8	5, 7	anbi12d 747	. . . . 5 |- ( m = M -> ( ( ( ( p ` 0 ) = -u pi /\ ( p ` m ) = pi ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) ) )
9	2, 8	rabeqbidv 3195	. . . 4 |- ( m = M -> { p e. ( ( -u pi [,] pi ) ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` m ) = pi ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } = { p e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
10		fourierdlem3.1	. . . 4 |- P = ( m e. NN |-> { p e. ( ( -u pi [,] pi ) ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` m ) = pi ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
11		ovex 6678	. . . . 5 |- ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) e. _V
12	11	rabex 4813	. . . 4 |- { p e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } e. _V
13	9, 10, 12	fvmpt 6282	. . 3 |- ( M e. NN -> ( P ` M ) = { p e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
14	13	eleq2d 2687	. 2 |- ( M e. NN -> ( Q e. ( P ` M ) <-> Q e. { p e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) )
15		fveq1 6190	. . . . . 6 |- ( p = Q -> ( p ` 0 ) = ( Q ` 0 ) )
16	15	eqeq1d 2624	. . . . 5 |- ( p = Q -> ( ( p ` 0 ) = -u pi <-> ( Q ` 0 ) = -u pi ) )
17		fveq1 6190	. . . . . 6 |- ( p = Q -> ( p ` M ) = ( Q ` M ) )
18	17	eqeq1d 2624	. . . . 5 |- ( p = Q -> ( ( p ` M ) = pi <-> ( Q ` M ) = pi ) )
19	16, 18	anbi12d 747	. . . 4 |- ( p = Q -> ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) <-> ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) ) )
20		fveq1 6190	. . . . . 6 |- ( p = Q -> ( p ` i ) = ( Q ` i ) )
21		fveq1 6190	. . . . . 6 |- ( p = Q -> ( p ` ( i + 1 ) ) = ( Q ` ( i + 1 ) ) )
22	20, 21	breq12d 4666	. . . . 5 |- ( p = Q -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
23	22	ralbidv 2986	. . . 4 |- ( p = Q -> ( A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
24	19, 23	anbi12d 747	. . 3 |- ( p = Q -> ( ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
25	24	elrab 3363	. 2 |- ( Q e. { p e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } <-> ( Q e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
26	14, 25	syl6bb 276	1 |- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( ( -u pi [,] pi ) ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   -ucneg 10267   NNcn 11020   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   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

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)