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Theorem eulerpartlemsv3 28125
Description: Lemma for eulerpart 28146. 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
eulerpartlemsv3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  ( 1 ... ( S `
 A ) ) ( ( A `  k )  x.  k
) )
Distinct variable groups:    f, k, A    R, f, k    S, k
Allowed substitution hint:    S( f)

Proof of Theorem eulerpartlemsv3
Dummy variable  t is distinct from all other variables.
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 28120 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 fzssuz 11736 . . . . 5  |-  ( 1 ... ( S `  A ) )  C_  ( ZZ>= `  1 )
5 nnuz 11129 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
64, 5sseqtr4i 3542 . . . 4  |-  ( 1 ... ( S `  A ) )  C_  NN
76a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  (
1 ... ( S `  A ) )  C_  NN )
81, 2eulerpartlemelr 28121 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
98simpld 459 . . . . . . 7  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
109adantr 465 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  A : NN --> NN0 )
117sselda 3509 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  NN )
1210, 11ffvelrnd 6033 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  NN0 )
1312nn0cnd 10866 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  CC )
1411nncnd 10564 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  CC )
1513, 14mulcld 9628 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
161, 2eulerpartlems 28124 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) )  -> 
( A `  t
)  =  0 )
1716ralrimiva 2881 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A. t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `  t
)  =  0 )
18 fveq2 5872 . . . . . . . . . 10  |-  ( k  =  t  ->  ( A `  k )  =  ( A `  t ) )
1918eqeq1d 2469 . . . . . . . . 9  |-  ( k  =  t  ->  (
( A `  k
)  =  0  <->  ( A `  t )  =  0 ) )
2019cbvralv 3093 . . . . . . . 8  |-  ( A. k  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `
 k )  =  0  <->  A. t  e.  (
ZZ>= `  ( ( S `
 A )  +  1 ) ) ( A `  t )  =  0 )
2117, 20sylibr 212 . . . . . . 7  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A. k  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `  k
)  =  0 )
221, 2eulerpartlemsf 28123 . . . . . . . . . 10  |-  S :
( ( NN0  ^m  NN )  i^i  R ) --> NN0
2322ffvelrni 6031 . . . . . . . . 9  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  e.  NN0 )
24 nndiffz1 27419 . . . . . . . . 9  |-  ( ( S `  A )  e.  NN0  ->  ( NN 
\  ( 1 ... ( S `  A
) ) )  =  ( ZZ>= `  ( ( S `  A )  +  1 ) ) )
2523, 24syl 16 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( NN  \  ( 1 ... ( S `  A
) ) )  =  ( ZZ>= `  ( ( S `  A )  +  1 ) ) )
2625raleqdv 3069 . . . . . . 7  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A. k  e.  ( NN  \  ( 1 ... ( S `  A
) ) ) ( A `  k )  =  0  <->  A. k  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `  k
)  =  0 ) )
2721, 26mpbird 232 . . . . . 6  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A. k  e.  ( NN  \  (
1 ... ( S `  A ) ) ) ( A `  k
)  =  0 )
2827r19.21bi 2836 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( A `  k
)  =  0 )
2928oveq1d 6310 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
30 simpr 461 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  ( NN 
\  ( 1 ... ( S `  A
) ) ) )
3130eldifad 3493 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  NN )
3231nncnd 10564 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  CC )
3332mul02d 9789 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( 0  x.  k
)  =  0 )
3429, 33eqtrd 2508 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
355eqimssi 3563 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3635a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
377, 15, 34, 36sumss 13526 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  sum_ k  e.  ( 1 ... ( S `  A )
) ( ( A `
 k )  x.  k )  =  sum_ k  e.  NN  (
( A `  k
)  x.  k ) )
383, 37eqtr4d 2511 1  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  ( 1 ... ( S `
 A ) ) ( ( A `  k )  x.  k
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817    \ cdif 3478    i^i cin 3480    C_ wss 3481    |-> cmpt 4511   `'ccnv 5004   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Fincfn 7528   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   NNcn 10548   NN0cn0 10807   ZZ>=cuz 11094   ...cfz 11684   sum_csu 13488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489
This theorem is referenced by:  eulerpartlemgc  28126
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