Theorem fourierdlem2 40326

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

Hypothesis

Ref	Expression
fourierdlem2.1	|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )

Assertion

Ref	Expression
fourierdlem2	|- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )

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

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

Proof of Theorem fourierdlem2

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( m = M -> ( 0 ... m ) = ( 0 ... M ) )
2	1	oveq2d 6666	. . . . 5 |- ( m = M -> ( RR ^m ( 0 ... m ) ) = ( RR ^m ( 0 ... M ) ) )
3		fveq2 6191	. . . . . . . 8 |- ( m = M -> ( p ` m ) = ( p ` M ) )
4	3	eqeq1d 2624	. . . . . . 7 |- ( m = M -> ( ( p ` m ) = B <-> ( p ` M ) = B ) )
5	4	anbi2d 740	. . . . . 6 |- ( m = M -> ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) <-> ( ( p ` 0 ) = A /\ ( p ` M ) = B ) ) )
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 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) ) )
9	2, 8	rabeqbidv 3195	. . . 4 |- ( m = M -> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } = { p e. ( RR ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
10		fourierdlem2.1	. . . 4 |- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
11		ovex 6678	. . . . 5 |- ( RR ^m ( 0 ... M ) ) e. _V
12	11	rabex 4813	. . . 4 |- { p e. ( RR ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ 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. ( RR ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ 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. ( RR ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ 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 ) = A <-> ( Q ` 0 ) = A ) )
17		fveq1 6190	. . . . . 6 |- ( p = Q -> ( p ` M ) = ( Q ` M ) )
18	17	eqeq1d 2624	. . . . 5 |- ( p = Q -> ( ( p ` M ) = B <-> ( Q ` M ) = B ) )
19	16, 18	anbi12d 747	. . . 4 |- ( p = Q -> ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) <-> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) ) )
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 ) = A /\ ( p ` M ) = B ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
25	24	elrab 3363	. 2 |- ( Q e. { p e. ( RR ^m ( 0 ... M ) ) | ( ( ( p ` 0 ) = A /\ ( p ` M ) = B ) /\ A. i e. ( 0..^ M ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ 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. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ 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   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   ...cfz 12326  ..^cfzo 12465

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:  fourierdlem11  40335  fourierdlem12  40336  fourierdlem13  40337  fourierdlem14  40338  fourierdlem15  40339  fourierdlem34  40358  fourierdlem37  40361  fourierdlem41  40365  fourierdlem48  40371  fourierdlem49  40372  fourierdlem50  40373  fourierdlem54  40377  fourierdlem63  40386  fourierdlem64  40387  fourierdlem65  40388  fourierdlem69  40392  fourierdlem70  40393  fourierdlem72  40395  fourierdlem74  40397  fourierdlem75  40398  fourierdlem76  40399  fourierdlem79  40402  fourierdlem81  40404  fourierdlem85  40408  fourierdlem88  40411  fourierdlem89  40412  fourierdlem90  40413  fourierdlem91  40414  fourierdlem92  40415  fourierdlem93  40416  fourierdlem94  40417  fourierdlem97  40420  fourierdlem102  40425  fourierdlem103  40426  fourierdlem104  40427  fourierdlem111  40434  fourierdlem113  40436  fourierdlem114  40437