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Theorem eulerpartlemgf 26926
Description: Lemma for eulerpart 26929: Images under  G have finite support. (Contributed by Thierry Arnoux, 29-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 ) ) ) ) ) )
Assertion
Ref Expression
eulerpartlemgf  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  e. 
Fin )
Distinct variable groups:    f, g,
k, n, o, x, y, z    f, r, A, o    o, F   
o, H, r    f, J    n, r, J, o, x, y    o, M, r    f, N, g, x    P, g, n    R, f, o    T, f, o
Allowed substitution hints:    A( x, y, z, g, k, n)    D( x, y, z, f, g, k, n, o, r)    P( x, y, z, f, k, o, r)    R( x, y, z, g, k, n, r)    T( x, y, z, 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)    J( z, g, k)    M( x, y, z, f, g, k, n)    N( y,
z, k, n, o, r)    O( x, y, z, f, g, k, n, o, r)

Proof of Theorem eulerpartlemgf
StepHypRef Expression
1 eulerpart.p . . . . . . 7  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
2 eulerpart.o . . . . . . 7  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
3 eulerpart.d . . . . . . 7  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
4 eulerpart.j . . . . . . 7  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
5 eulerpart.f . . . . . . 7  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
6 eulerpart.h . . . . . . 7  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
7 eulerpart.m . . . . . . 7  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
8 eulerpart.r . . . . . . 7  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
9 eulerpart.t . . . . . . 7  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
10 eulerpart.g . . . . . . 7  |-  G  =  ( o  e.  ( T  i^i  R ) 
|->  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( o  |`  J ) ) ) ) ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemgv 26920 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1211cnveqd 5126 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  `' ( G `  A )  =  `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1312imaeq1d 5279 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } ) )
14 nnex 10442 . . . . 5  |-  NN  e.  _V
15 imassrn 5291 . . . . . 6  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  ran  F
164, 5oddpwdc 26901 . . . . . . 7  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
17 f1of 5752 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  F :
( J  X.  NN0 )
--> NN )
18 frn 5676 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) --> NN  ->  ran  F 
C_  NN )
1916, 17, 18mp2b 10 . . . . . 6  |-  ran  F  C_  NN
2015, 19sstri 3476 . . . . 5  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  NN
21 indpi1 26643 . . . . 5  |-  ( ( NN  e.  _V  /\  ( F " ( M `
 (bits  o.  ( A  |`  J ) ) ) )  C_  NN )  ->  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) " { 1 } )  =  ( F "
( M `  (bits  o.  ( A  |`  J ) ) ) ) )
2214, 20, 21mp2an 672 . . . 4  |-  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )
2313, 22syl6eq 2511 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
24 inss2 3682 . . . . 5  |-  ( ~P ( J  X.  NN0 )  i^i  Fin )  C_  Fin
251, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 26922 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  H )
261, 2, 3, 4, 5, 6, 7eulerpartlem1 26914 . . . . . . . 8  |-  M : H
-1-1-onto-> ( ~P ( J  X.  NN0 )  i^i  Fin )
27 f1of 5752 . . . . . . . 8  |-  ( M : H -1-1-onto-> ( ~P ( J  X.  NN0 )  i^i 
Fin )  ->  M : H --> ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
2826, 27ax-mp 5 . . . . . . 7  |-  M : H
--> ( ~P ( J  X.  NN0 )  i^i 
Fin )
2928ffvelrni 5954 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e.  H  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
3025, 29syl 16 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
3124, 30sseldi 3465 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )
32 ffun 5672 . . . . . 6  |-  ( F : ( J  X.  NN0 ) --> NN  ->  Fun  F )
3316, 17, 32mp2b 10 . . . . 5  |-  Fun  F
34 imafi 7718 . . . . 5  |-  ( ( Fun  F  /\  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e.  Fin )
3533, 34mpan 670 . . . 4  |-  ( ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e.  Fin )
3631, 35syl 16 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e. 
Fin )
3723, 36eqeltrd 2542 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  e.  Fin )
381, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartgbij 26919 . . . . . . . 8  |-  G :
( T  i^i  R
)
-1-1-onto-> ( ( { 0 ,  1 }  ^m  NN )  i^i  R )
39 f1of 5752 . . . . . . . 8  |-  ( G : ( T  i^i  R ) -1-1-onto-> ( ( { 0 ,  1 }  ^m  NN )  i^i  R )  ->  G : ( T  i^i  R ) --> ( ( { 0 ,  1 }  ^m  NN )  i^i  R ) )
4038, 39ax-mp 5 . . . . . . 7  |-  G :
( T  i^i  R
) --> ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )
4140ffvelrni 5954 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i  R ) )
42 elin 3650 . . . . . . 7  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  <->  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  /\  ( G `  A )  e.  R
) )
4342simplbi 460 . . . . . 6  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  ->  ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN ) )
44 elmapi 7347 . . . . . 6  |-  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  ->  ( G `
 A ) : NN --> { 0 ,  1 } )
4541, 43, 443syl 20 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A ) : NN --> { 0 ,  1 } )
46 ffun 5672 . . . . 5  |-  ( ( G `  A ) : NN --> { 0 ,  1 }  ->  Fun  ( G `  A
) )
4745, 46syl 16 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( G `
 A ) )
48 ssv 3487 . . . . 5  |-  NN0  C_  _V
49 dfn2 10706 . . . . . 6  |-  NN  =  ( NN0  \  { 0 } )
50 ssdif 3602 . . . . . 6  |-  ( NN0  C_  _V  ->  ( NN0  \  { 0 } ) 
C_  ( _V  \  { 0 } ) )
5149, 50syl5eqss 3511 . . . . 5  |-  ( NN0  C_  _V  ->  NN  C_  ( _V  \  { 0 } ) )
5248, 51ax-mp 5 . . . 4  |-  NN  C_  ( _V  \  { 0 } )
53 sspreima 26133 . . . 4  |-  ( ( Fun  ( G `  A )  /\  NN  C_  ( _V  \  {
0 } ) )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
5447, 52, 53sylancl 662 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
55 fvex 5812 . . . . 5  |-  ( G `
 A )  e. 
_V
56 0nn0 10708 . . . . 5  |-  0  e.  NN0
57 suppimacnv 6814 . . . . 5  |-  ( ( ( G `  A
)  e.  _V  /\  0  e.  NN0 )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" ( _V  \  { 0 } ) ) )
5855, 56, 57mp2an 672 . . . 4  |-  ( ( G `  A ) supp  0 )  =  ( `' ( G `  A ) " ( _V  \  { 0 } ) )
59 ax-1ne0 9465 . . . . . . . . . 10  |-  1  =/=  0
6059necomi 2722 . . . . . . . . 9  |-  0  =/=  1
61 difprsn1 4121 . . . . . . . . 9  |-  ( 0  =/=  1  ->  ( { 0 ,  1 }  \  { 0 } )  =  {
1 } )
6260, 61ax-mp 5 . . . . . . . 8  |-  ( { 0 ,  1 } 
\  { 0 } )  =  { 1 }
6362eqcomi 2467 . . . . . . 7  |-  { 1 }  =  ( { 0 ,  1 } 
\  { 0 } )
6463ffs2 26199 . . . . . 6  |-  ( ( NN  e.  _V  /\  0  e.  NN0  /\  ( G `  A ) : NN --> { 0 ,  1 } )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" { 1 } ) )
6514, 56, 64mp3an12 1305 . . . . 5  |-  ( ( G `  A ) : NN --> { 0 ,  1 }  ->  ( ( G `  A
) supp  0 )  =  ( `' ( G `
 A ) " { 1 } ) )
6645, 65syl 16 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( ( G `  A ) supp  0 )  =  ( `' ( G `  A ) " {
1 } ) )
6758, 66syl5eqr 2509 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" ( _V  \  { 0 } ) )  =  ( `' ( G `  A
) " { 1 } ) )
6854, 67sseqtrd 3503 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " {
1 } ) )
69 ssfi 7647 . 2  |-  ( ( ( `' ( G `
 A ) " { 1 } )  e.  Fin  /\  ( `' ( G `  A ) " NN )  C_  ( `' ( G `  A )
" { 1 } ) )  ->  ( `' ( G `  A ) " NN )  e.  Fin )
7037, 68, 69syl2anc 661 1  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078    \ cdif 3436    i^i cin 3438    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   {csn 3988   {cpr 3990   class class class wbr 4403   {copab 4460    |-> cmpt 4461    X. cxp 4949   `'ccnv 4950   ran crn 4952    |` cres 4953   "cima 4954    o. ccom 4955   Fun wfun 5523   -->wf 5525   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   supp csupp 6803    ^m cmap 7327   Fincfn 7423   0cc0 9396   1c1 9397    x. cmul 9401    <_ cle 9533   NNcn 10436   2c2 10485   NN0cn0 10693   ^cexp 11985   sum_csu 13284    || cdivides 13656  bitscbits 13736  𝟭cind 26632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-ac2 8746  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-disj 4374  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7805  df-oi 7838  df-card 8223  df-acn 8226  df-ac 8400  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-dvds 13657  df-bits 13739  df-ind 26633
This theorem is referenced by:  eulerpartlemgs2  26927
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