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Theorem eulerpartlemsv3 28567
Description: Lemma for eulerpart 28588. 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 28562 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 fzssuz 11728 . . . . 5  |-  ( 1 ... ( S `  A ) )  C_  ( ZZ>= `  1 )
5 nnuz 11117 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
64, 5sseqtr4i 3522 . . . 4  |-  ( 1 ... ( S `  A ) )  C_  NN
76a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  (
1 ... ( S `  A ) )  C_  NN )
81, 2eulerpartlemelr 28563 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
98simpld 457 . . . . . . 7  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
109adantr 463 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  A : NN --> NN0 )
117sselda 3489 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  NN )
1210, 11ffvelrnd 6008 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  NN0 )
1312nn0cnd 10850 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  CC )
1411nncnd 10547 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  CC )
1513, 14mulcld 9605 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
161, 2eulerpartlems 28566 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) )  -> 
( A `  t
)  =  0 )
1716ralrimiva 2868 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A. t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `  t
)  =  0 )
18 fveq2 5848 . . . . . . . . . 10  |-  ( k  =  t  ->  ( A `  k )  =  ( A `  t ) )
1918eqeq1d 2456 . . . . . . . . 9  |-  ( k  =  t  ->  (
( A `  k
)  =  0  <->  ( A `  t )  =  0 ) )
2019cbvralv 3081 . . . . . . . 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 28565 . . . . . . . . . 10  |-  S :
( ( NN0  ^m  NN )  i^i  R ) --> NN0
2322ffvelrni 6006 . . . . . . . . 9  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  e.  NN0 )
24 nndiffz1 27833 . . . . . . . . 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 3057 . . . . . . 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 2823 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( A `  k
)  =  0 )
2928oveq1d 6285 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
30 simpr 459 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  ( NN 
\  ( 1 ... ( S `  A
) ) ) )
3130eldifad 3473 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  NN )
3231nncnd 10547 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  CC )
3332mul02d 9767 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( 0  x.  k
)  =  0 )
3429, 33eqtrd 2495 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
355eqimssi 3543 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3635a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
377, 15, 34, 36sumss 13631 . 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 2498 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 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804    \ cdif 3458    i^i cin 3460    C_ wss 3461    |-> cmpt 4497   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Fincfn 7509   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NNcn 10531   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675   sum_csu 13593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594
This theorem is referenced by:  eulerpartlemgc  28568
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