Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemgv Structured version   Visualization version   Unicode version

Theorem eulerpartlemgv 29279
Description: Lemma for eulerpart 29288: value of the function  G. (Contributed by Thierry Arnoux, 13-Nov-2017.)
Hypotheses
Ref Expression
eulerpart.p  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
eulerpart.o  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
eulerpart.d  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
eulerpart.j  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
eulerpart.f  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
eulerpart.h  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
eulerpart.m  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
eulerpart.r  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
eulerpart.t  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
eulerpart.g  |-  G  =  ( o  e.  ( T  i^i  R ) 
|->  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( o  |`  J ) ) ) ) ) )
Assertion
Ref Expression
eulerpartlemgv  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
Distinct variable groups:    A, o    o, F    o, J    o, M    R, o    T, o
Allowed substitution hints:    A( x, y, z, f, g, k, n, r)    D( x, y, z, f, g, k, n, o, r)    P( x, y, z, f, g, k, n, o, r)    R( x, y, z, f, g, k, n, r)    T( x, y, z, f, g, k, n, r)    F( x, y, z, f, g, k, n, r)    G( x, y, z, f, g, k, n, o, r)    H( x, y, z, f, g, k, n, o, r)    J( x, y, z, f, g, k, n, r)    M( x, y, z, f, g, k, n, r)    N( x, y, z, f, g, k, n, o, r)    O( x, y, z, f, g, k, n, o, r)

Proof of Theorem eulerpartlemgv
StepHypRef Expression
1 reseq1 5105 . . . . . 6  |-  ( o  =  A  ->  (
o  |`  J )  =  ( A  |`  J ) )
21coeq2d 5002 . . . . 5  |-  ( o  =  A  ->  (bits  o.  ( o  |`  J ) )  =  (bits  o.  ( A  |`  J ) ) )
32fveq2d 5883 . . . 4  |-  ( o  =  A  ->  ( M `  (bits  o.  (
o  |`  J ) ) )  =  ( M `
 (bits  o.  ( A  |`  J ) ) ) )
43imaeq2d 5174 . . 3  |-  ( o  =  A  ->  ( F " ( M `  (bits  o.  ( o  |`  J ) ) ) )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
54fveq2d 5883 . 2  |-  ( o  =  A  ->  (
(𝟭 `  NN ) `  ( F " ( M `
 (bits  o.  (
o  |`  J ) ) ) ) )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
6 eulerpart.g . 2  |-  G  =  ( o  e.  ( T  i^i  R ) 
|->  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( o  |`  J ) ) ) ) ) )
7 fvex 5889 . 2  |-  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )  e.  _V
85, 6, 7fvmpt 5963 1  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   {crab 2760    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   class class class wbr 4395   {copab 4453    |-> cmpt 4454   `'ccnv 4838    |` cres 4841   "cima 4842    o. ccom 4843   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   supp csupp 6933    ^m cmap 7490   Fincfn 7587   1c1 9558    x. cmul 9562    <_ cle 9694   NNcn 10631   2c2 10681   NN0cn0 10893   ^cexp 12310   sum_csu 13829    || cdvds 14382  bitscbits 14471  𝟭cind 28906
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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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-iota 5553  df-fun 5591  df-fv 5597
This theorem is referenced by:  eulerpartlemgvv  29282  eulerpartlemgf  29285  eulerpartlemn  29287
  Copyright terms: Public domain W3C validator