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Theorem eulerpartlemgf 28705
Description: Lemma for eulerpart 28708: 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 28699 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1211cnveqd 5118 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  `' ( G `  A )  =  `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1312imaeq1d 5275 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } ) )
14 nnex 10500 . . . . 5  |-  NN  e.  _V
15 imassrn 5287 . . . . . 6  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  ran  F
164, 5oddpwdc 28680 . . . . . . 7  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
17 f1of 5753 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  F :
( J  X.  NN0 )
--> NN )
18 frn 5674 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) --> NN  ->  ran  F 
C_  NN )
1916, 17, 18mp2b 10 . . . . . 6  |-  ran  F  C_  NN
2015, 19sstri 3448 . . . . 5  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  NN
21 indpi1 28350 . . . . 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 670 . . . 4  |-  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )
2313, 22syl6eq 2457 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
24 ffun 5670 . . . . 5  |-  ( F : ( J  X.  NN0 ) --> NN  ->  Fun  F )
2516, 17, 24mp2b 10 . . . 4  |-  Fun  F
26 inss2 3657 . . . . 5  |-  ( ~P ( J  X.  NN0 )  i^i  Fin )  C_  Fin
271, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 28701 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  H )
281, 2, 3, 4, 5, 6, 7eulerpartlem1 28693 . . . . . . . 8  |-  M : H
-1-1-onto-> ( ~P ( J  X.  NN0 )  i^i  Fin )
29 f1of 5753 . . . . . . . 8  |-  ( M : H -1-1-onto-> ( ~P ( J  X.  NN0 )  i^i 
Fin )  ->  M : H --> ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
3028, 29ax-mp 5 . . . . . . 7  |-  M : H
--> ( ~P ( J  X.  NN0 )  i^i 
Fin )
3130ffvelrni 5962 . . . . . 6  |-  ( (bits 
o.  ( A  |`  J ) )  e.  H  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
3227, 31syl 17 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  ( ~P ( J  X.  NN0 )  i^i 
Fin ) )
3326, 32sseldi 3437 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )
34 imafi 7765 . . . 4  |-  ( ( Fun  F  /\  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e.  Fin )
3525, 33, 34sylancr 661 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e. 
Fin )
3623, 35eqeltrd 2488 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  e.  Fin )
371, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartgbij 28698 . . . . . . . 8  |-  G :
( T  i^i  R
)
-1-1-onto-> ( ( { 0 ,  1 }  ^m  NN )  i^i  R )
38 f1of 5753 . . . . . . . 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 ) )
3937, 38ax-mp 5 . . . . . . 7  |-  G :
( T  i^i  R
) --> ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )
4039ffvelrni 5962 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i  R ) )
41 elin 3623 . . . . . . 7  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  <->  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  /\  ( G `  A )  e.  R
) )
4241simplbi 458 . . . . . 6  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  ->  ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN ) )
43 elmapi 7396 . . . . . 6  |-  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  ->  ( G `
 A ) : NN --> { 0 ,  1 } )
4440, 42, 433syl 20 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A ) : NN --> { 0 ,  1 } )
45 ffun 5670 . . . . 5  |-  ( ( G `  A ) : NN --> { 0 ,  1 }  ->  Fun  ( G `  A
) )
4644, 45syl 17 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( G `
 A ) )
47 ssv 3459 . . . . 5  |-  NN0  C_  _V
48 dfn2 10767 . . . . . 6  |-  NN  =  ( NN0  \  { 0 } )
49 ssdif 3575 . . . . . 6  |-  ( NN0  C_  _V  ->  ( NN0  \  { 0 } ) 
C_  ( _V  \  { 0 } ) )
5048, 49syl5eqss 3483 . . . . 5  |-  ( NN0  C_  _V  ->  NN  C_  ( _V  \  { 0 } ) )
5147, 50ax-mp 5 . . . 4  |-  NN  C_  ( _V  \  { 0 } )
52 sspreima 27809 . . . 4  |-  ( ( Fun  ( G `  A )  /\  NN  C_  ( _V  \  {
0 } ) )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
5346, 51, 52sylancl 660 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
54 fvex 5813 . . . . 5  |-  ( G `
 A )  e. 
_V
55 0nn0 10769 . . . . 5  |-  0  e.  NN0
56 suppimacnv 6865 . . . . 5  |-  ( ( ( G `  A
)  e.  _V  /\  0  e.  NN0 )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" ( _V  \  { 0 } ) ) )
5754, 55, 56mp2an 670 . . . 4  |-  ( ( G `  A ) supp  0 )  =  ( `' ( G `  A ) " ( _V  \  { 0 } ) )
58 0ne1 10562 . . . . . . . . 9  |-  0  =/=  1
59 difprsn1 4105 . . . . . . . . 9  |-  ( 0  =/=  1  ->  ( { 0 ,  1 }  \  { 0 } )  =  {
1 } )
6058, 59ax-mp 5 . . . . . . . 8  |-  ( { 0 ,  1 } 
\  { 0 } )  =  { 1 }
6160eqcomi 2413 . . . . . . 7  |-  { 1 }  =  ( { 0 ,  1 } 
\  { 0 } )
6261ffs2 27879 . . . . . 6  |-  ( ( NN  e.  _V  /\  0  e.  NN0  /\  ( G `  A ) : NN --> { 0 ,  1 } )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" { 1 } ) )
6314, 55, 62mp3an12 1314 . . . . 5  |-  ( ( G `  A ) : NN --> { 0 ,  1 }  ->  ( ( G `  A
) supp  0 )  =  ( `' ( G `
 A ) " { 1 } ) )
6444, 63syl 17 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( ( G `  A ) supp  0 )  =  ( `' ( G `  A ) " {
1 } ) )
6557, 64syl5eqr 2455 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" ( _V  \  { 0 } ) )  =  ( `' ( G `  A
) " { 1 } ) )
6653, 65sseqtrd 3475 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " {
1 } ) )
67 ssfi 7693 . 2  |-  ( ( ( `' ( G `
 A ) " { 1 } )  e.  Fin  /\  ( `' ( G `  A ) " NN )  C_  ( `' ( G `  A )
" { 1 } ) )  ->  ( `' ( G `  A ) " NN )  e.  Fin )
6836, 66, 67syl2anc 659 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 367    = wceq 1403    e. wcel 1840   {cab 2385    =/= wne 2596   A.wral 2751   {crab 2755   _Vcvv 3056    \ cdif 3408    i^i cin 3410    C_ wss 3411   (/)c0 3735   ~Pcpw 3952   {csn 3969   {cpr 3971   class class class wbr 4392   {copab 4449    |-> cmpt 4450    X. cxp 4938   `'ccnv 4939   ran crn 4941    |` cres 4942   "cima 4943    o. ccom 4944   Fun wfun 5517   -->wf 5519   -1-1-onto->wf1o 5522   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   supp csupp 6854    ^m cmap 7375   Fincfn 7472   0cc0 9440   1c1 9441    x. cmul 9445    <_ cle 9577   NNcn 10494   2c2 10544   NN0cn0 10754   ^cexp 12118   sum_csu 13562    || cdvds 14085  bitscbits 14168  𝟭cind 28339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-ac2 8793  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-disj 4364  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-sup 7853  df-oi 7887  df-card 8270  df-acn 8273  df-ac 8447  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563  df-dvds 14086  df-bits 14171  df-ind 28340
This theorem is referenced by:  eulerpartlemgs2  28706
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