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Theorem fourierdlem14 38095
Description: Given the partition  V,  Q is the partition shifted to the left by  X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem14.1  |-  ( ph  ->  A  e.  RR )
fourierdlem14.2  |-  ( ph  ->  B  e.  RR )
fourierdlem14.x  |-  ( ph  ->  X  e.  RR )
fourierdlem14.p  |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  ( A  +  X
)  /\  ( p `  m )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `  i
)  <  ( p `  ( i  +  1 ) ) ) } )
fourierdlem14.o  |-  O  =  ( 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 ) ) ) } )
fourierdlem14.m  |-  ( ph  ->  M  e.  NN )
fourierdlem14.v  |-  ( ph  ->  V  e.  ( P `
 M ) )
fourierdlem14.q  |-  Q  =  ( i  e.  ( 0 ... M ) 
|->  ( ( V `  i )  -  X
) )
Assertion
Ref Expression
fourierdlem14  |-  ( ph  ->  Q  e.  ( O `
 M ) )
Distinct variable groups:    A, m, p    B, m, p    i, M, m, p    Q, i, p    i, V, p   
i, X, m, p    ph, i
Allowed substitution hints:    ph( m, p)    A( i)    B( i)    P( i, m, p)    Q( m)    O( i, m, p)    V( m)

Proof of Theorem fourierdlem14
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fourierdlem14.v . . . . . . . . . 10  |-  ( ph  ->  V  e.  ( P `
 M ) )
2 fourierdlem14.m . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
3 fourierdlem14.p . . . . . . . . . . . 12  |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  ( A  +  X
)  /\  ( p `  m )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `  i
)  <  ( p `  ( i  +  1 ) ) ) } )
43fourierdlem2 38083 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  ( V  e.  ( P `  M )  <->  ( V  e.  ( RR  ^m  (
0 ... M ) )  /\  ( ( ( V `  0 )  =  ( A  +  X )  /\  ( V `  M )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ M ) ( V `
 i )  < 
( V `  (
i  +  1 ) ) ) ) ) )
52, 4syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( V  e.  ( P `  M )  <-> 
( V  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( V `  0 )  =  ( A  +  X )  /\  ( V `  M )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ M ) ( V `
 i )  < 
( V `  (
i  +  1 ) ) ) ) ) )
61, 5mpbid 215 . . . . . . . . 9  |-  ( ph  ->  ( V  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( V `  0 )  =  ( A  +  X )  /\  ( V `  M )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ M ) ( V `
 i )  < 
( V `  (
i  +  1 ) ) ) ) )
76simpld 466 . . . . . . . 8  |-  ( ph  ->  V  e.  ( RR 
^m  ( 0 ... M ) ) )
8 elmapi 7511 . . . . . . . 8  |-  ( V  e.  ( RR  ^m  ( 0 ... M
) )  ->  V : ( 0 ... M ) --> RR )
97, 8syl 17 . . . . . . 7  |-  ( ph  ->  V : ( 0 ... M ) --> RR )
109fnvinran 37398 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( V `  i )  e.  RR )
11 fourierdlem14.x . . . . . . 7  |-  ( ph  ->  X  e.  RR )
1211adantr 472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  X  e.  RR )
1310, 12resubcld 10068 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  (
( V `  i
)  -  X )  e.  RR )
14 fourierdlem14.q . . . . 5  |-  Q  =  ( i  e.  ( 0 ... M ) 
|->  ( ( V `  i )  -  X
) )
1513, 14fmptd 6061 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
16 reex 9648 . . . . . 6  |-  RR  e.  _V
1716a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
18 ovex 6336 . . . . . 6  |-  ( 0 ... M )  e. 
_V
1918a1i 11 . . . . 5  |-  ( ph  ->  ( 0 ... M
)  e.  _V )
2017, 19elmapd 7504 . . . 4  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  <-> 
Q : ( 0 ... M ) --> RR ) )
2115, 20mpbird 240 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
2214a1i 11 . . . . . 6  |-  ( ph  ->  Q  =  ( i  e.  ( 0 ... M )  |->  ( ( V `  i )  -  X ) ) )
23 fveq2 5879 . . . . . . . 8  |-  ( i  =  0  ->  ( V `  i )  =  ( V ` 
0 ) )
2423oveq1d 6323 . . . . . . 7  |-  ( i  =  0  ->  (
( V `  i
)  -  X )  =  ( ( V `
 0 )  -  X ) )
2524adantl 473 . . . . . 6  |-  ( (
ph  /\  i  = 
0 )  ->  (
( V `  i
)  -  X )  =  ( ( V `
 0 )  -  X ) )
26 0zd 10973 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
272nnzd 11062 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
2826, 27, 263jca 1210 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  e.  ZZ )
)
29 0le0 10721 . . . . . . . . 9  |-  0  <_  0
3029a1i 11 . . . . . . . 8  |-  ( ph  ->  0  <_  0 )
31 0red 9662 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
322nnred 10646 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
332nngt0d 10675 . . . . . . . . 9  |-  ( ph  ->  0  <  M )
3431, 32, 33ltled 9800 . . . . . . . 8  |-  ( ph  ->  0  <_  M )
3528, 30, 34jca32 544 . . . . . . 7  |-  ( ph  ->  ( ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  e.  ZZ )  /\  (
0  <_  0  /\  0  <_  M ) ) )
36 elfz2 11817 . . . . . . 7  |-  ( 0  e.  ( 0 ... M )  <->  ( (
0  e.  ZZ  /\  M  e.  ZZ  /\  0  e.  ZZ )  /\  (
0  <_  0  /\  0  <_  M ) ) )
3735, 36sylibr 217 . . . . . 6  |-  ( ph  ->  0  e.  ( 0 ... M ) )
389, 37ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( V `  0
)  e.  RR )
3938, 11resubcld 10068 . . . . . 6  |-  ( ph  ->  ( ( V ` 
0 )  -  X
)  e.  RR )
4022, 25, 37, 39fvmptd 5969 . . . . 5  |-  ( ph  ->  ( Q `  0
)  =  ( ( V `  0 )  -  X ) )
416simprd 470 . . . . . . . 8  |-  ( ph  ->  ( ( ( V `
 0 )  =  ( A  +  X
)  /\  ( V `  M )  =  ( B  +  X ) )  /\  A. i  e.  ( 0..^ M ) ( V `  i
)  <  ( V `  ( i  +  1 ) ) ) )
4241simpld 466 . . . . . . 7  |-  ( ph  ->  ( ( V ` 
0 )  =  ( A  +  X )  /\  ( V `  M )  =  ( B  +  X ) ) )
4342simpld 466 . . . . . 6  |-  ( ph  ->  ( V `  0
)  =  ( A  +  X ) )
4443oveq1d 6323 . . . . 5  |-  ( ph  ->  ( ( V ` 
0 )  -  X
)  =  ( ( A  +  X )  -  X ) )
45 fourierdlem14.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
4645recnd 9687 . . . . . 6  |-  ( ph  ->  A  e.  CC )
4711recnd 9687 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4846, 47pncand 10006 . . . . 5  |-  ( ph  ->  ( ( A  +  X )  -  X
)  =  A )
4940, 44, 483eqtrd 2509 . . . 4  |-  ( ph  ->  ( Q `  0
)  =  A )
50 fveq2 5879 . . . . . . . 8  |-  ( i  =  M  ->  ( V `  i )  =  ( V `  M ) )
5150oveq1d 6323 . . . . . . 7  |-  ( i  =  M  ->  (
( V `  i
)  -  X )  =  ( ( V `
 M )  -  X ) )
5251adantl 473 . . . . . 6  |-  ( (
ph  /\  i  =  M )  ->  (
( V `  i
)  -  X )  =  ( ( V `
 M )  -  X ) )
5326, 27, 273jca 1210 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  M  e.  ZZ  /\  M  e.  ZZ )
)
5432leidd 10201 . . . . . . . 8  |-  ( ph  ->  M  <_  M )
5553, 34, 54jca32 544 . . . . . . 7  |-  ( ph  ->  ( ( 0  e.  ZZ  /\  M  e.  ZZ  /\  M  e.  ZZ )  /\  (
0  <_  M  /\  M  <_  M ) ) )
56 elfz2 11817 . . . . . . 7  |-  ( M  e.  ( 0 ... M )  <->  ( (
0  e.  ZZ  /\  M  e.  ZZ  /\  M  e.  ZZ )  /\  (
0  <_  M  /\  M  <_  M ) ) )
5755, 56sylibr 217 . . . . . 6  |-  ( ph  ->  M  e.  ( 0 ... M ) )
589, 57ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( V `  M
)  e.  RR )
5958, 11resubcld 10068 . . . . . 6  |-  ( ph  ->  ( ( V `  M )  -  X
)  e.  RR )
6022, 52, 57, 59fvmptd 5969 . . . . 5  |-  ( ph  ->  ( Q `  M
)  =  ( ( V `  M )  -  X ) )
6142simprd 470 . . . . . 6  |-  ( ph  ->  ( V `  M
)  =  ( B  +  X ) )
6261oveq1d 6323 . . . . 5  |-  ( ph  ->  ( ( V `  M )  -  X
)  =  ( ( B  +  X )  -  X ) )
63 fourierdlem14.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
6463recnd 9687 . . . . . 6  |-  ( ph  ->  B  e.  CC )
6564, 47pncand 10006 . . . . 5  |-  ( ph  ->  ( ( B  +  X )  -  X
)  =  B )
6660, 62, 653eqtrd 2509 . . . 4  |-  ( ph  ->  ( Q `  M
)  =  B )
6749, 66jca 541 . . 3  |-  ( ph  ->  ( ( Q ` 
0 )  =  A  /\  ( Q `  M )  =  B ) )
68 elfzofz 11962 . . . . . . 7  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
6968, 10sylan2 482 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( V `  i )  e.  RR )
709adantr 472 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  V : ( 0 ... M ) --> RR )
71 fzofzp1 12037 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
7271adantl 473 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  +  1 )  e.  ( 0 ... M ) )
7370, 72ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( V `  ( i  +  1 ) )  e.  RR )
7411adantr 472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  X  e.  RR )
7541simprd 470 . . . . . . 7  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( V `  i )  <  ( V `  ( i  +  1 ) ) )
7675r19.21bi 2776 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( V `  i )  <  ( V `  ( i  +  1 ) ) )
7769, 73, 74, 76ltsub1dd 10246 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( V `
 i )  -  X )  <  (
( V `  (
i  +  1 ) )  -  X ) )
7868adantl 473 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0 ... M ) )
7968, 13sylan2 482 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( V `
 i )  -  X )  e.  RR )
8014fvmpt2 5972 . . . . . 6  |-  ( ( i  e.  ( 0 ... M )  /\  ( ( V `  i )  -  X
)  e.  RR )  ->  ( Q `  i )  =  ( ( V `  i
)  -  X ) )
8178, 79, 80syl2anc 673 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  =  ( ( V `  i
)  -  X ) )
82 fveq2 5879 . . . . . . . . . 10  |-  ( i  =  j  ->  ( V `  i )  =  ( V `  j ) )
8382oveq1d 6323 . . . . . . . . 9  |-  ( i  =  j  ->  (
( V `  i
)  -  X )  =  ( ( V `
 j )  -  X ) )
8483cbvmptv 4488 . . . . . . . 8  |-  ( i  e.  ( 0 ... M )  |->  ( ( V `  i )  -  X ) )  =  ( j  e.  ( 0 ... M
)  |->  ( ( V `
 j )  -  X ) )
8514, 84eqtri 2493 . . . . . . 7  |-  Q  =  ( j  e.  ( 0 ... M ) 
|->  ( ( V `  j )  -  X
) )
8685a1i 11 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q  =  ( j  e.  ( 0 ... M )  |->  ( ( V `  j
)  -  X ) ) )
87 fveq2 5879 . . . . . . . 8  |-  ( j  =  ( i  +  1 )  ->  ( V `  j )  =  ( V `  ( i  +  1 ) ) )
8887oveq1d 6323 . . . . . . 7  |-  ( j  =  ( i  +  1 )  ->  (
( V `  j
)  -  X )  =  ( ( V `
 ( i  +  1 ) )  -  X ) )
8988adantl 473 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  j  =  ( i  +  1 ) )  ->  (
( V `  j
)  -  X )  =  ( ( V `
 ( i  +  1 ) )  -  X ) )
9073, 74resubcld 10068 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( V `
 ( i  +  1 ) )  -  X )  e.  RR )
9186, 89, 72, 90fvmptd 5969 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  =  ( ( V `  (
i  +  1 ) )  -  X ) )
9277, 81, 913brtr4d 4426 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
9392ralrimiva 2809 . . 3  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
9421, 67, 93jca32 544 . 2  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) )
95 fourierdlem14.o . . . 4  |-  O  =  ( 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 ) ) ) } )
9695fourierdlem2 38083 . . 3  |-  ( M  e.  NN  ->  ( Q  e.  ( O `  M )  <->  ( Q  e.  ( RR  ^m  (
0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
972, 96syl 17 . 2  |-  ( ph  ->  ( Q  e.  ( O `  M )  <-> 
( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
9894, 97mpbird 240 1  |-  ( ph  ->  Q  e.  ( O `
 M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   ZZcz 10961   ...cfz 11810  ..^cfzo 11942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943
This theorem is referenced by:  fourierdlem74  38156  fourierdlem75  38157  fourierdlem84  38166  fourierdlem85  38167  fourierdlem88  38170  fourierdlem103  38185  fourierdlem104  38186
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