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Theorem eulerpartlemsv2 26693
Description: Lemma for eulerpart 26717. 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 26691 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 cnvimass 5184 . . . 4  |-  ( `' A " NN ) 
C_  dom  A
51, 2eulerpartlemelr 26692 . . . . . 6  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
65simpld 459 . . . . 5  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
7 fdm 5558 . . . . 5  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
86, 7syl 16 . . . 4  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  dom  A  =  NN )
94, 8syl5sseq 3399 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( `' A " NN ) 
C_  NN )
106adantr 465 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
119sselda 3351 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
1210, 11ffvelrnd 5839 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
1311nnnn0d 10628 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
1412, 13nn0mulcld 10633 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1514nn0cnd 10630 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
16 simpr 461 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1716eldifad 3335 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1816eldifbd 3336 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
196adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
20 ffn 5554 . . . . . . . . . 10  |-  ( A : NN --> NN0  ->  A  Fn  NN )
21 elpreima 5818 . . . . . . . . . 10  |-  ( A  Fn  NN  ->  (
k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
2219, 20, 213syl 20 . . . . . . . . 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 300 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
24 imnan 422 . . . . . . . 8  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2523, 24sylibr 212 . . . . . . 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 5839 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
28 elnn0 10573 . . . . . . 7  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2927, 28sylib 196 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
30 orel1 382 . . . . . 6  |-  ( -.  ( A `  k
)  e.  NN  ->  ( ( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 )  ->  ( A `  k )  =  0 ) )
3126, 29, 30sylc 60 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  =  0 )
3231oveq1d 6101 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
3317nncnd 10330 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3433mul02d 9559 . . . 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 10888 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
3736eqimssi 3405 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3837a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
399, 15, 35, 38sumss 13193 . 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 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2424    \ cdif 3320    i^i cin 3322    C_ wss 3323    e. cmpt 4345   `'ccnv 4834   dom cdm 4835   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   Fincfn 7302   0cc0 9274   1c1 9275    x. cmul 9279   NNcn 10314   NN0cn0 10571   ZZ>=cuz 10853   sum_csu 13155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156
This theorem is referenced by:  eulerpartlemsf  26694  eulerpartlemgs2  26715
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