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Theorem fourierdlem2 37912
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
StepHypRef Expression
1 oveq2 6314 . . . . . 6  |-  ( m  =  M  ->  (
0 ... m )  =  ( 0 ... M
) )
21oveq2d 6322 . . . . 5  |-  ( m  =  M  ->  ( RR  ^m  ( 0 ... m ) )  =  ( RR  ^m  (
0 ... M ) ) )
3 fveq2 5882 . . . . . . . 8  |-  ( m  =  M  ->  (
p `  m )  =  ( p `  M ) )
43eqeq1d 2424 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  m
)  =  B  <->  ( p `  M )  =  B ) )
54anbi2d 708 . . . . . 6  |-  ( m  =  M  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  m )  =  B )  <->  ( ( p `
 0 )  =  A  /\  ( p `
 M )  =  B ) ) )
6 oveq2 6314 . . . . . . 7  |-  ( m  =  M  ->  (
0..^ m )  =  ( 0..^ M ) )
76raleqdv 3028 . . . . . 6  |-  ( m  =  M  ->  ( A. i  e.  (
0..^ m ) ( p `  i )  <  ( p `  ( i  +  1 ) )  <->  A. i  e.  ( 0..^ M ) ( p `  i
)  <  ( p `  ( i  +  1 ) ) ) )
85, 7anbi12d 715 . . . . 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 ) ) ) ) )
92, 8rabeqbidv 3075 . . . 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 6334 . . . . 5  |-  ( RR 
^m  ( 0 ... M ) )  e. 
_V
1211rabex 4575 . . . 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
139, 10, 12fvmpt 5965 . . 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 ) ) ) } )
1413eleq2d 2492 . 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 5881 . . . . . 6  |-  ( p  =  Q  ->  (
p `  0 )  =  ( Q ` 
0 ) )
1615eqeq1d 2424 . . . . 5  |-  ( p  =  Q  ->  (
( p `  0
)  =  A  <->  ( Q `  0 )  =  A ) )
17 fveq1 5881 . . . . . 6  |-  ( p  =  Q  ->  (
p `  M )  =  ( Q `  M ) )
1817eqeq1d 2424 . . . . 5  |-  ( p  =  Q  ->  (
( p `  M
)  =  B  <->  ( Q `  M )  =  B ) )
1916, 18anbi12d 715 . . . 4  |-  ( p  =  Q  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  M )  =  B )  <->  ( ( Q `
 0 )  =  A  /\  ( Q `
 M )  =  B ) ) )
20 fveq1 5881 . . . . . 6  |-  ( p  =  Q  ->  (
p `  i )  =  ( Q `  i ) )
21 fveq1 5881 . . . . . 6  |-  ( p  =  Q  ->  (
p `  ( i  +  1 ) )  =  ( Q `  ( i  +  1 ) ) )
2220, 21breq12d 4436 . . . . 5  |-  ( p  =  Q  ->  (
( p `  i
)  <  ( p `  ( i  +  1 ) )  <->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) ) )
2322ralbidv 2861 . . . 4  |-  ( p  =  Q  ->  ( A. i  e.  (
0..^ M ) ( p `  i )  <  ( p `  ( i  +  1 ) )  <->  A. i  e.  ( 0..^ M ) ( Q `  i
)  <  ( Q `  ( i  +  1 ) ) ) )
2419, 23anbi12d 715 . . 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 ) ) ) ) )
2524elrab 3228 . 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 ) ) ) ) )
2614, 25syl6bb 264 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 ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775   class class class wbr 4423    |-> cmpt 4482   ` cfv 5601  (class class class)co 6306    ^m cmap 7484   RRcr 9546   0cc0 9547   1c1 9548    + caddc 9550    < clt 9683   NNcn 10617   ...cfz 11792  ..^cfzo 11923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6309
This theorem is referenced by:  fourierdlem11  37921  fourierdlem12  37922  fourierdlem13  37923  fourierdlem14  37924  fourierdlem15  37925  fourierdlem34  37945  fourierdlem37  37948  fourierdlem41  37952  fourierdlem48  37959  fourierdlem49  37960  fourierdlem50  37961  fourierdlem54  37965  fourierdlem63  37974  fourierdlem64  37975  fourierdlem65  37976  fourierdlem69  37980  fourierdlem70  37981  fourierdlem72  37983  fourierdlem74  37985  fourierdlem75  37986  fourierdlem76  37987  fourierdlem79  37990  fourierdlem81  37992  fourierdlem85  37996  fourierdlem88  37999  fourierdlem89  38000  fourierdlem90  38001  fourierdlem91  38002  fourierdlem92  38003  fourierdlem93  38004  fourierdlem94  38005  fourierdlem97  38008  fourierdlem102  38013  fourierdlem103  38014  fourierdlem104  38015  fourierdlem111  38022  fourierdlem113  38024  fourierdlem114  38025
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