Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem34 Structured version   Unicode version

Theorem fourierdlem34 32089
Description: A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem34.p  |-  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 ) ) ) } )
fourierdlem34.m  |-  ( ph  ->  M  e.  NN )
fourierdlem34.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
Assertion
Ref Expression
fourierdlem34  |-  ( ph  ->  Q : ( 0 ... M ) -1-1-> RR )
Distinct variable groups:    A, m, p    B, m, p    i, M, m, p    Q, i, p    ph, i
Allowed substitution hints:    ph( m, p)    A( i)    B( i)    P( i, m, p)    Q( m)

Proof of Theorem fourierdlem34
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem34.q . . . . 5  |-  ( ph  ->  Q  e.  ( P `
 M ) )
2 fourierdlem34.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
3 fourierdlem34.p . . . . . . 7  |-  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 ) ) ) } )
43fourierdlem2 32057 . . . . . 6  |-  ( 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 ) ) ) ) ) )
52, 4syl 16 . . . . 5  |-  ( ph  ->  ( 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 ) ) ) ) ) )
61, 5mpbid 210 . . . 4  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) )
76simpld 457 . . 3  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
8 elmapi 7359 . . 3  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  ->  Q : ( 0 ... M ) --> RR )
97, 8syl 16 . 2  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
10 simplr 753 . . . . . 6  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  ( Q `
 i )  =  ( Q `  j
) )  /\  -.  i  =  j )  ->  ( Q `  i
)  =  ( Q `
 j ) )
119ffvelrnda 5933 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
1211ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  ( Q `  i )  e.  RR )
139ffvelrnda 5933 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( 0 ... M
) )  ->  ( Q `  k )  e.  RR )
1413adant423 31592 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  i  <  j
)  /\  k  e.  ( 0 ... M
) )  ->  ( Q `  k )  e.  RR )
1514adantllr 716 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  /\  k  e.  ( 0 ... M
) )  ->  ( Q `  k )  e.  RR )
16 eleq1 2454 . . . . . . . . . . . . . . . . 17  |-  ( i  =  k  ->  (
i  e.  ( 0..^ M )  <->  k  e.  ( 0..^ M ) ) )
1716anbi2d 701 . . . . . . . . . . . . . . . 16  |-  ( i  =  k  ->  (
( ph  /\  i  e.  ( 0..^ M ) )  <->  ( ph  /\  k  e.  ( 0..^ M ) ) ) )
18 fveq2 5774 . . . . . . . . . . . . . . . . 17  |-  ( i  =  k  ->  ( Q `  i )  =  ( Q `  k ) )
19 oveq1 6203 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  k  ->  (
i  +  1 )  =  ( k  +  1 ) )
2019fveq2d 5778 . . . . . . . . . . . . . . . . 17  |-  ( i  =  k  ->  ( Q `  ( i  +  1 ) )  =  ( Q `  ( k  +  1 ) ) )
2118, 20breq12d 4380 . . . . . . . . . . . . . . . 16  |-  ( i  =  k  ->  (
( Q `  i
)  <  ( Q `  ( i  +  1 ) )  <->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) ) )
2217, 21imbi12d 318 . . . . . . . . . . . . . . 15  |-  ( i  =  k  ->  (
( ( ph  /\  i  e.  ( 0..^ M ) )  -> 
( Q `  i
)  <  ( Q `  ( i  +  1 ) ) )  <->  ( ( ph  /\  k  e.  ( 0..^ M ) )  ->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) ) ) )
236simprrd 756 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
2423r19.21bi 2751 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
2522, 24chvarv 2021 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( 0..^ M ) )  ->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) )
2625adant423 31592 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  i  <  j
)  /\  k  e.  ( 0..^ M ) )  ->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) )
2726adantllr 716 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  /\  k  e.  ( 0..^ M ) )  ->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) )
28 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  i  e.  ( 0 ... M
) )
29 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  j  e.  ( 0 ... M
) )
30 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  i  <  j )
3115, 27, 28, 29, 30monoords 31662 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  ( Q `  i )  <  ( Q `  j
) )
3212, 31ltned 9632 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  ( Q `  i )  =/=  ( Q `  j
) )
3332neneqd 2584 . . . . . . . . 9  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  i  < 
j )  ->  -.  ( Q `  i )  =  ( Q `  j ) )
3433adantlr 712 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  i  <  j )  ->  -.  ( Q `  i
)  =  ( Q `
 j ) )
35 simpll 751 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) ) )
36 elfzelz 11609 . . . . . . . . . . . 12  |-  ( j  e.  ( 0 ... M )  ->  j  e.  ZZ )
3736zred 10884 . . . . . . . . . . 11  |-  ( j  e.  ( 0 ... M )  ->  j  e.  RR )
3837ad3antlr 728 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  j  e.  RR )
39 elfzelz 11609 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... M )  ->  i  e.  ZZ )
4039zred 10884 . . . . . . . . . . 11  |-  ( i  e.  ( 0 ... M )  ->  i  e.  RR )
4140ad4antlr 730 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  i  e.  RR )
42 neqne 31601 . . . . . . . . . . . 12  |-  ( -.  i  =  j  -> 
i  =/=  j )
4342necomd 2653 . . . . . . . . . . 11  |-  ( -.  i  =  j  -> 
j  =/=  i )
4443ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  j  =/=  i
)
45 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  -.  i  <  j )
4638, 41, 44, 45lttri5d 31665 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  j  <  i
)
479ffvelrnda 5933 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 0 ... M
) )  ->  ( Q `  j )  e.  RR )
4847adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  j  e.  ( 0 ... M
) )  /\  j  <  i )  ->  ( Q `  j )  e.  RR )
4948adantllr 716 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  ( Q `  j )  e.  RR )
50 simp-4l 765 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  /\  k  e.  ( 0 ... M
) )  ->  ph )
5150, 13sylancom 665 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  /\  k  e.  ( 0 ... M
) )  ->  ( Q `  k )  e.  RR )
52 simp-4l 765 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  /\  k  e.  ( 0..^ M ) )  ->  ph )
5352, 25sylancom 665 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  /\  k  e.  ( 0..^ M ) )  ->  ( Q `  k )  <  ( Q `  ( k  +  1 ) ) )
54 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  j  e.  ( 0 ... M
) )
55 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  i  e.  ( 0 ... M
) )
56 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  j  <  i )
5751, 53, 54, 55, 56monoords 31662 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  ( Q `  j )  <  ( Q `  i
) )
5849, 57gtned 9631 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  ( Q `  i )  =/=  ( Q `  j
) )
5958neneqd 2584 . . . . . . . . 9  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  j  < 
i )  ->  -.  ( Q `  i )  =  ( Q `  j ) )
6035, 46, 59syl2anc 659 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  /\  -.  i  <  j )  ->  -.  ( Q `  i )  =  ( Q `  j ) )
6134, 60pm2.61dan 789 . . . . . . 7  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  -.  i  =  j )  ->  -.  ( Q `  i
)  =  ( Q `
 j ) )
6261adantlr 712 . . . . . 6  |-  ( ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  ( Q `
 i )  =  ( Q `  j
) )  /\  -.  i  =  j )  ->  -.  ( Q `  i )  =  ( Q `  j ) )
6310, 62condan 792 . . . . 5  |-  ( ( ( ( ph  /\  i  e.  ( 0 ... M ) )  /\  j  e.  ( 0 ... M ) )  /\  ( Q `
 i )  =  ( Q `  j
) )  ->  i  =  j )
6463ex 432 . . . 4  |-  ( ( ( ph  /\  i  e.  ( 0 ... M
) )  /\  j  e.  ( 0 ... M
) )  ->  (
( Q `  i
)  =  ( Q `
 j )  -> 
i  =  j ) )
6564ralrimiva 2796 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  A. j  e.  ( 0 ... M
) ( ( Q `
 i )  =  ( Q `  j
)  ->  i  =  j ) )
6665ralrimiva 2796 . 2  |-  ( ph  ->  A. i  e.  ( 0 ... M ) A. j  e.  ( 0 ... M ) ( ( Q `  i )  =  ( Q `  j )  ->  i  =  j ) )
67 dff13 6067 . 2  |-  ( Q : ( 0 ... M ) -1-1-> RR  <->  ( Q : ( 0 ... M ) --> RR  /\  A. i  e.  ( 0 ... M ) A. j  e.  ( 0 ... M ) ( ( Q `  i
)  =  ( Q `
 j )  -> 
i  =  j ) ) )
689, 66, 67sylanbrc 662 1  |-  ( ph  ->  Q : ( 0 ... M ) -1-1-> RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   {crab 2736   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   -1-1->wf1 5493   ` cfv 5496  (class class class)co 6196    ^m cmap 7338   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    < clt 9539   NNcn 10452   ...cfz 11593  ..^cfzo 11717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718
This theorem is referenced by:  fourierdlem50  32105
  Copyright terms: Public domain W3C validator