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Theorem eulerpartlemgf 29262
Description: Lemma for eulerpart 29265: 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 29256 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  =  ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1211cnveqd 5032 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  `' ( G `  A )  =  `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) ) )
1312imaeq1d 5189 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } ) )
14 nnex 10648 . . . . 5  |-  NN  e.  _V
15 imassrn 5201 . . . . . 6  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  ran  F
164, 5oddpwdc 29237 . . . . . . 7  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
17 f1of 5841 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  F :
( J  X.  NN0 )
--> NN )
18 frn 5762 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) --> NN  ->  ran  F 
C_  NN )
1916, 17, 18mp2b 10 . . . . . 6  |-  ran  F  C_  NN
2015, 19sstri 3453 . . . . 5  |-  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )  C_  NN
21 indpi1 28894 . . . . 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 683 . . . 4  |-  ( `' ( (𝟭 `  NN ) `  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) )
2313, 22syl6eq 2512 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  =  ( F
" ( M `  (bits  o.  ( A  |`  J ) ) ) ) )
24 ffun 5758 . . . . 5  |-  ( F : ( J  X.  NN0 ) --> NN  ->  Fun  F )
2516, 17, 24mp2b 10 . . . 4  |-  Fun  F
26 inss2 3665 . . . . 5  |-  ( ~P ( J  X.  NN0 )  i^i  Fin )  C_  Fin
271, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 29258 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  (bits  o.  ( A  |`  J ) )  e.  H )
281, 2, 3, 4, 5, 6, 7eulerpartlem1 29250 . . . . . . . 8  |-  M : H
-1-1-onto-> ( ~P ( J  X.  NN0 )  i^i  Fin )
29 f1of 5841 . . . . . . . 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 6049 . . . . . 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 3442 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )
34 imafi 7898 . . . 4  |-  ( ( Fun  F  /\  ( M `  (bits  o.  ( A  |`  J ) ) )  e.  Fin )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e.  Fin )
3525, 33, 34sylancr 674 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( F " ( M `  (bits  o.  ( A  |`  J ) ) ) )  e. 
Fin )
3623, 35eqeltrd 2540 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" { 1 } )  e.  Fin )
371, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartgbij 29255 . . . . . . . 8  |-  G :
( T  i^i  R
)
-1-1-onto-> ( ( { 0 ,  1 }  ^m  NN )  i^i  R )
38 f1of 5841 . . . . . . . 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 6049 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i  R ) )
41 elin 3629 . . . . . . 7  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  <->  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  /\  ( G `  A )  e.  R
) )
4241simplbi 466 . . . . . 6  |-  ( ( G `  A )  e.  ( ( { 0 ,  1 }  ^m  NN )  i^i 
R )  ->  ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN ) )
43 elmapi 7524 . . . . . 6  |-  ( ( G `  A )  e.  ( { 0 ,  1 }  ^m  NN )  ->  ( G `
 A ) : NN --> { 0 ,  1 } )
4440, 42, 433syl 18 . . . . 5  |-  ( A  e.  ( T  i^i  R )  ->  ( G `  A ) : NN --> { 0 ,  1 } )
45 ffun 5758 . . . . 5  |-  ( ( G `  A ) : NN --> { 0 ,  1 }  ->  Fun  ( G `  A
) )
4644, 45syl 17 . . . 4  |-  ( A  e.  ( T  i^i  R )  ->  Fun  ( G `
 A ) )
47 ssv 3464 . . . . 5  |-  NN0  C_  _V
48 dfn2 10916 . . . . . 6  |-  NN  =  ( NN0  \  { 0 } )
49 ssdif 3580 . . . . . 6  |-  ( NN0  C_  _V  ->  ( NN0  \  { 0 } ) 
C_  ( _V  \  { 0 } ) )
5048, 49syl5eqss 3488 . . . . 5  |-  ( NN0  C_  _V  ->  NN  C_  ( _V  \  { 0 } ) )
5147, 50ax-mp 5 . . . 4  |-  NN  C_  ( _V  \  { 0 } )
52 sspreima 28299 . . . 4  |-  ( ( Fun  ( G `  A )  /\  NN  C_  ( _V  \  {
0 } ) )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
5346, 51, 52sylancl 673 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " ( _V  \  { 0 } ) ) )
54 fvex 5902 . . . . 5  |-  ( G `
 A )  e. 
_V
55 0nn0 10918 . . . . 5  |-  0  e.  NN0
56 suppimacnv 6957 . . . . 5  |-  ( ( ( G `  A
)  e.  _V  /\  0  e.  NN0 )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" ( _V  \  { 0 } ) ) )
5754, 55, 56mp2an 683 . . . 4  |-  ( ( G `  A ) supp  0 )  =  ( `' ( G `  A ) " ( _V  \  { 0 } ) )
58 0ne1 10710 . . . . . . . . 9  |-  0  =/=  1
59 difprsn1 4121 . . . . . . . . 9  |-  ( 0  =/=  1  ->  ( { 0 ,  1 }  \  { 0 } )  =  {
1 } )
6058, 59ax-mp 5 . . . . . . . 8  |-  ( { 0 ,  1 } 
\  { 0 } )  =  { 1 }
6160eqcomi 2471 . . . . . . 7  |-  { 1 }  =  ( { 0 ,  1 } 
\  { 0 } )
6261ffs2 28365 . . . . . 6  |-  ( ( NN  e.  _V  /\  0  e.  NN0  /\  ( G `  A ) : NN --> { 0 ,  1 } )  -> 
( ( G `  A ) supp  0 )  =  ( `' ( G `  A )
" { 1 } ) )
6314, 55, 62mp3an12 1363 . . . . 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 2510 . . 3  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" ( _V  \  { 0 } ) )  =  ( `' ( G `  A
) " { 1 } ) )
6653, 65sseqtrd 3480 . 2  |-  ( A  e.  ( T  i^i  R )  ->  ( `' ( G `  A )
" NN )  C_  ( `' ( G `  A ) " {
1 } ) )
67 ssfi 7823 . 2  |-  ( ( ( `' ( G `
 A ) " { 1 } )  e.  Fin  /\  ( `' ( G `  A ) " NN )  C_  ( `' ( G `  A )
" { 1 } ) )  ->  ( `' ( G `  A ) " NN )  e.  Fin )
6836, 66, 67syl2anc 671 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 375    = wceq 1455    e. wcel 1898   {cab 2448    =/= wne 2633   A.wral 2749   {crab 2753   _Vcvv 3057    \ cdif 3413    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {csn 3980   {cpr 3982   class class class wbr 4418   {copab 4476    |-> cmpt 4477    X. cxp 4854   `'ccnv 4855   ran crn 4857    |` cres 4858   "cima 4859    o. ccom 4860   Fun wfun 5599   -->wf 5601   -1-1-onto->wf1o 5604   ` cfv 5605  (class class class)co 6320    |-> cmpt2 6322   supp csupp 6946    ^m cmap 7503   Fincfn 7600   0cc0 9570   1c1 9571    x. cmul 9575    <_ cle 9707   NNcn 10642   2c2 10692   NN0cn0 10903   ^cexp 12310   sum_csu 13807    || cdvds 14360  bitscbits 14447  𝟭cind 28883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-ac2 8924  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-disj 4390  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-supp 6947  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-2o 7214  df-oadd 7217  df-er 7394  df-map 7505  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fsupp 7915  df-sup 7987  df-inf 7988  df-oi 8056  df-card 8404  df-acn 8407  df-ac 8578  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-fz 11820  df-fzo 11953  df-fl 12066  df-mod 12135  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-clim 13607  df-sum 13808  df-dvds 14361  df-bits 14450  df-ind 28884
This theorem is referenced by:  eulerpartlemgs2  29263
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