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Theorem eulerpartlemsv2 29240
Description: Lemma for eulerpart 29264. 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 29238 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( S `  A )  =  sum_ k  e.  NN  ( ( A `  k )  x.  k
) )
4 cnvimass 5207 . . . 4  |-  ( `' A " NN ) 
C_  dom  A
51, 2eulerpartlemelr 29239 . . . . . 6  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
65simpld 465 . . . . 5  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
7 fdm 5756 . . . . 5  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
86, 7syl 17 . . . 4  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  dom  A  =  NN )
94, 8syl5sseq 3492 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( `' A " NN ) 
C_  NN )
106adantr 471 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
119sselda 3444 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
1210, 11ffvelrnd 6046 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
1311nnnn0d 10954 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
1412, 13nn0mulcld 10959 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1514nn0cnd 10956 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
16 simpr 467 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1716eldifad 3428 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1816eldifbd 3429 . . . . . . . . 9  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
196adantr 471 . . . . . . . . . 10  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
20 ffn 5751 . . . . . . . . . 10  |-  ( A : NN --> NN0  ->  A  Fn  NN )
21 elpreima 6025 . . . . . . . . . 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 306 . . . . . . . 8  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
24 imnan 428 . . . . . . . 8  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2523, 24sylibr 217 . . . . . . 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 6046 . . . . . . 7  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
28 elnn0 10900 . . . . . . 7  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2927, 28sylib 201 . . . . . 6  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
30 orel1 388 . . . . . 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 6330 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
3317nncnd 10653 . . . . 5  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3433mul02d 9857 . . . 4  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( 0  x.  k
)  =  0 )
3532, 34eqtrd 2496 . . 3  |-  ( ( A  e.  ( ( NN0  ^m  NN )  i^i  R )  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
36 nnuz 11223 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
3736eqimssi 3498 . . . 4  |-  NN  C_  ( ZZ>= `  1 )
3837a1i 11 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  NN  C_  ( ZZ>= `  1 )
)
399, 15, 35, 38sumss 13839 . 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 2499 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 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448    \ cdif 3413    i^i cin 3415    C_ wss 3416    |-> cmpt 4475   `'ccnv 4852   dom cdm 4853   "cima 4856    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315    ^m cmap 7498   Fincfn 7595   0cc0 9565   1c1 9566    x. cmul 9570   NNcn 10637   NN0cn0 10898   ZZ>=cuz 11188   sum_csu 13801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fz 11814  df-fzo 11947  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-sum 13802
This theorem is referenced by:  eulerpartlemsf  29241  eulerpartlemgs2  29262
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