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Theorem eulerpartlemgu 27984
Description: Lemma for eulerpart 27989: Rewriting the  U set for an odd partition Note that interestingly, this proof reuses marypha2lem2 7896. (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 2467 . . . 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 7896 . . 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 27976 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
1413simp1bi 1011 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
15 elmapi 7440 . . . . . . . . 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 5733 . . . . . . 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 3718 . . . . . . . . 9  |-  ( ( `' A " NN )  i^i  J )  C_  ( `' A " NN )
21 cnvimass 5357 . . . . . . . . . 10  |-  ( `' A " NN ) 
C_  dom  A
22 fdm 5735 . . . . . . . . . . 11  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
2316, 22syl 16 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  ->  dom  A  =  NN )
2421, 23syl5sseq 3552 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  C_  NN )
2520, 24syl5ss 3515 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  ( ( `' A " NN )  i^i  J )  C_  NN )
2625sselda 3504 . . . . . . 7  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( ( `' A " NN )  i^i  J ) )  ->  t  e.  NN )
2723eleq2d 2537 . . . . . . . 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 5943 . . . . . 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 5023 . . . 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 4347 . . 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 2523 . 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 2520 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   U_ciun 4325   class class class wbr 4447   {copab 4504    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   supp csupp 6901    ^m cmap 7420   Fincfn 7516   1c1 9493    x. cmul 9497    <_ cle 9629   NNcn 10536   2c2 10585   NN0cn0 10795   ^cexp 12134   sum_csu 13471    || cdivides 13847  bitscbits 13928  𝟭cind 27692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422
This theorem is referenced by: (None)
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