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Theorem eulerpartlemsv2 29008
Description: Lemma for eulerpart 29032. Value of the sum of a finite partition  A (Contributed by Thierry Arnoux, 19-Aug-2018.)
Hypotheses
Ref Expression
eulerpartlems.r  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
eulerpartlems.s  |-  S  =  ( f  e.  ( ( NN0  ^m  NN )  i^i  R )  |->  sum_ k  e.  NN  (
( f `  k
)  x.  k ) )
Assertion
Ref Expression
eulerpartlemsv2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
Distinct variable groups:    f, k, A    R, f, k
Allowed substitution hints:    S( f, k)

Proof of Theorem eulerpartlemsv2
StepHypRef Expression
1 eulerpartlems.r . . 3  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
2 eulerpartlems.s . . 3  |-  S  =  ( f  e.  ( ( NN0  ^m  NN )  i^i  R )  |->  sum_ k  e.  NN  (
( f `  k
)  x.  k ) )
31, 2eulerpartlemsv1 29006 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 cnvimass 5208 . . . 4  |-  ( `' A " NN ) 
C_  dom  A
51, 2eulerpartlemelr 29007 . . . . . 6  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
65simpld 460 . . . . 5  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
7 fdm 5750 . . . . 5  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
86, 7syl 17 . . . 4  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  dom  A  =  NN )
94, 8syl5sseq 3518 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( `' A " NN ) 
C_  NN )
106adantr 466 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
119sselda 3470 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
1210, 11ffvelrnd 6038 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
1311nnnn0d 10925 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
1412, 13nn0mulcld 10930 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1514nn0cnd 10927 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
16 simpr 462 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1716eldifad 3454 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1816eldifbd 3455 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
196adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
20 ffn 5746 . . . . . . . . . 10  |-  ( A : NN --> NN0  ->  A  Fn  NN )
21 elpreima 6017 . . . . . . . . . 10  |-  ( A  Fn  NN  ->  (
k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
2219, 20, 213syl 18 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
2318, 22mtbid 301 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
24 imnan 423 . . . . . . . 8  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2523, 24sylibr 215 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( k  e.  NN  ->  -.  ( A `  k )  e.  NN ) )
2617, 25mpd 15 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( A `  k
)  e.  NN )
2719, 17ffvelrnd 6038 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
28 elnn0 10871 . . . . . . 7  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2927, 28sylib 199 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
30 orel1 383 . . . . . 6  |-  ( -.  ( A `  k
)  e.  NN  ->  ( ( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 )  ->  ( A `  k )  =  0 ) )
3126, 29, 30sylc 62 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  =  0 )
3231oveq1d 6320 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
3317nncnd 10625 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3433mul02d 9830 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( 0  x.  k
)  =  0 )
3532, 34eqtrd 2470 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
36 nnuz 11194 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
3736eqimssi 3524 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3837a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
399, 15, 35, 38sumss 13768 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  sum_ k  e.  ( `' A " NN ) ( ( A `
 k )  x.  k )  =  sum_ k  e.  NN  (
( A `  k
)  x.  k ) )
403, 39eqtr4d 2473 1  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414    \ cdif 3439    i^i cin 3441    C_ wss 3442    |-> cmpt 4484   `'ccnv 4853   dom cdm 4854   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   Fincfn 7577   0cc0 9538   1c1 9539    x. cmul 9543   NNcn 10609   NN0cn0 10869   ZZ>=cuz 11159   sum_csu 13730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731
This theorem is referenced by:  eulerpartlemsf  29009  eulerpartlemgs2  29030
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