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Theorem eulerpartlemsv3 26744
Description: Lemma for eulerpart 26765. 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 26739 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 fzssuz 11499 . . . . 5  |-  ( 1 ... ( S `  A ) )  C_  ( ZZ>= `  1 )
5 nnuz 10896 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
64, 5sseqtr4i 3389 . . . 4  |-  ( 1 ... ( S `  A ) )  C_  NN
76a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  (
1 ... ( S `  A ) )  C_  NN )
81, 2eulerpartlemelr 26740 . . . . . . . 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 3356 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  NN )
1210, 11ffvelrnd 5844 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  NN0 )
1312nn0cnd 10638 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( A `  k )  e.  CC )
1411nncnd 10338 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  k  e.  CC )
1513, 14mulcld 9406 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( 1 ... ( S `  A ) ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
161, 2eulerpartlems 26743 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) )  -> 
( A `  t
)  =  0 )
1716ralrimiva 2799 . . . . . . . 8  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A. t  e.  ( ZZ>= `  ( ( S `  A )  +  1 ) ) ( A `  t
)  =  0 )
18 fveq2 5691 . . . . . . . . . 10  |-  ( k  =  t  ->  ( A `  k )  =  ( A `  t ) )
1918eqeq1d 2451 . . . . . . . . 9  |-  ( k  =  t  ->  (
( A `  k
)  =  0  <->  ( A `  t )  =  0 ) )
2019cbvralv 2947 . . . . . . . 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 26742 . . . . . . . . . 10  |-  S :
( ( NN0  ^m  NN )  i^i  R ) --> NN0
2322ffvelrni 5842 . . . . . . . . 9  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  e.  NN0 )
24 nndiffz1 26075 . . . . . . . . 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 2923 . . . . . . 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 2814 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( A `  k
)  =  0 )
2928oveq1d 6106 . . . 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 3340 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  NN )
3231nncnd 10338 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
k  e.  CC )
3332mul02d 9567 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( 0  x.  k
)  =  0 )
3429, 33eqtrd 2475 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( 1 ... ( S `  A )
) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
355eqimssi 3410 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3635a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
377, 15, 34, 36sumss 13201 . 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 2478 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 1369    e. wcel 1756   {cab 2429   A.wral 2715    \ cdif 3325    i^i cin 3327    C_ wss 3328    e. cmpt 4350   `'ccnv 4839   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Fincfn 7310   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437   sum_csu 13163
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164
This theorem is referenced by:  eulerpartlemgc  26745
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