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

Theorem eulerpartlemgu 26782
Description: Lemma for eulerpart 26787: Rewriting the  U set for an odd partition Note that interestingly, this proof reuses marypha2lem2 7707. (Contributed by Thierry Arnoux, 10-Aug-2018.)
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 ) ) ) ) ) )
eulerpartlemgh.1  |-  U  = 
U_ t  e.  ( ( `' A " NN )  i^i  J ) ( { t }  X.  (bits `  ( A `  t )
) )
Assertion
Ref Expression
eulerpartlemgu  |-  ( A  e.  ( T  i^i  R )  ->  U  =  { <. t ,  n >.  |  ( t  e.  ( ( `' A " NN )  i^i  J
)  /\  n  e.  ( (bits  o.  A
) `  t )
) } )
Distinct variable groups:    z, t    f, g, k, n, t, A    f, J, n, t    f, N, k, n, t    n, O, t    P, g, k    R, f, k, n, t    T, n, t
Allowed substitution hints:    A( x, y, z, o, r)    D( x, y, z, t, f, g, k, n, o, r)    P( x, y, z, t, f, n, o, r)    R( x, y, z, g, o, r)    T( x, y, z, f, g, k, o, r)    U( x, y, z, t, f, g, k, n, o, r)    F( x, y, z, t, f, g, k, n, o, r)    G( x, y, z, t, f, g, k, n, o, r)    H( x, y, z, t, f, g, k, n, o, r)    J( x, y, z, g, k, o, r)    M( x, y, z, t, f, g, k, n, o, r)    N( x, y, z, g, o, r)    O( x, y, z, f, g, k, o, r)

Proof of Theorem eulerpartlemgu
StepHypRef Expression
1 eulerpartlemgh.1 . 2  |-  U  = 
U_ t  e.  ( ( `' A " NN )  i^i  J ) ( { t }  X.  (bits `  ( A `  t )
) )
2 eqid 2443 . . . 4  |-  U_ t  e.  ( ( `' A " NN )  i^i  J
) ( { t }  X.  ( (bits 
o.  A ) `  t ) )  = 
U_ t  e.  ( ( `' A " NN )  i^i  J ) ( { t }  X.  ( (bits  o.  A ) `  t
) )
32marypha2lem2 7707 . . 3  |-  U_ t  e.  ( ( `' A " NN )  i^i  J
) ( { t }  X.  ( (bits 
o.  A ) `  t ) )  =  { <. t ,  n >.  |  ( t  e.  ( ( `' A " NN )  i^i  J
)  /\  n  e.  ( (bits  o.  A
) `  t )
) }
4 eulerpart.p . . . . . . . . . . 11  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
5 eulerpart.o . . . . . . . . . . 11  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
6 eulerpart.d . . . . . . . . . . 11  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
7 eulerpart.j . . . . . . . . . . 11  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
8 eulerpart.f . . . . . . . . . . 11  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
9 eulerpart.h . . . . . . . . . . 11  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
10 eulerpart.m . . . . . . . . . . 11  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
11 eulerpart.r . . . . . . . . . . 11  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
12 eulerpart.t . . . . . . . . . . 11  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
134, 5, 6, 7, 8, 9, 10, 11, 12eulerpartlemt0 26774 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413simp1bi 1003 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
15 elmapi 7255 . . . . . . . . 9  |-  ( A  e.  ( NN0  ^m  NN )  ->  A : NN
--> NN0 )
1614, 15syl 16 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
1716adantr 465 . . . . . . 7  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  A : NN --> NN0 )
18 ffun 5582 . . . . . . 7  |-  ( A : NN --> NN0  ->  Fun 
A )
1917, 18syl 16 . . . . . 6  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  Fun  A )
20 inss1 3591 . . . . . . . . 9  |-  ( ( `' A " NN )  i^i  J )  C_  ( `' A " NN )
21 cnvimass 5210 . . . . . . . . . 10  |-  ( `' A " NN ) 
C_  dom  A
22 fdm 5584 . . . . . . . . . . 11  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
2316, 22syl 16 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  ->  dom  A  =  NN )
2421, 23syl5sseq 3425 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  C_  NN )
2520, 24syl5ss 3388 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " NN )  i^i  J )  C_  NN )
2625sselda 3377 . . . . . . 7  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  t  e.  NN )
2723eleq2d 2510 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( t  e.  dom  A  <->  t  e.  NN ) )
2827adantr 465 . . . . . . 7  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  ( t  e. 
dom  A  <->  t  e.  NN ) )
2926, 28mpbird 232 . . . . . 6  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  t  e.  dom  A )
30 fvco 5788 . . . . . 6  |-  ( ( Fun  A  /\  t  e.  dom  A )  -> 
( (bits  o.  A
) `  t )  =  (bits `  ( A `  t ) ) )
3119, 29, 30syl2anc 661 . . . . 5  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  ( (bits  o.  A ) `  t
)  =  (bits `  ( A `  t ) ) )
3231xpeq2d 4885 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  ( { t }  X.  ( (bits 
o.  A ) `  t ) )  =  ( { t }  X.  (bits `  ( A `  t )
) ) )
3332iuneq2dv 4213 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  U_ t  e.  ( ( `' A " NN )  i^i  J
) ( { t }  X.  ( (bits 
o.  A ) `  t ) )  = 
U_ t  e.  ( ( `' A " NN )  i^i  J ) ( { t }  X.  (bits `  ( A `  t )
) ) )
343, 33syl5reqr 2490 . 2  |-  ( A  e.  ( T  i^i  R )  ->  U_ t  e.  ( ( `' A " NN )  i^i  J
) ( { t }  X.  (bits `  ( A `  t ) ) )  =  { <. t ,  n >.  |  ( t  e.  ( ( `' A " NN )  i^i  J )  /\  n  e.  ( (bits  o.  A ) `
 t ) ) } )
351, 34syl5eq 2487 1  |-  ( A  e.  ( T  i^i  R )  ->  U  =  { <. t ,  n >.  |  ( t  e.  ( ( `' A " NN )  i^i  J
)  /\  n  e.  ( (bits  o.  A
) `  t )
) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2736   {crab 2740    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   U_ciun 4192   class class class wbr 4313   {copab 4370    e. cmpt 4371    X. cxp 4859   `'ccnv 4860   dom cdm 4861    |` cres 4863   "cima 4864    o. ccom 4865   Fun wfun 5433   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   supp csupp 6711    ^m cmap 7235   Fincfn 7331   1c1 9304    x. cmul 9308    <_ cle 9440   NNcn 10343   2c2 10392   NN0cn0 10600   ^cexp 11886   sum_csu 13184    || cdivides 13556  bitscbits 13636  𝟭cind 26489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-map 7237
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator