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Theorem eulerpartlemv 29245
Description: Lemma for eulerpart 29263. (Contributed by Thierry Arnoux, 19-Aug-2018.)
Hypothesis
Ref Expression
eulerpart.p  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
Assertion
Ref Expression
eulerpartlemv  |-  ( A  e.  P  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k
)  x.  k )  =  N ) )
Distinct variable groups:    f, k, A    f, N, k    P, k
Allowed substitution hint:    P( f)

Proof of Theorem eulerpartlemv
StepHypRef Expression
1 eulerpart.p . . 3  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
21eulerpartleme 29244 . 2  |-  ( A  e.  P  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N ) )
3 cnvimass 5206 . . . . . . . . 9  |-  ( `' A " NN ) 
C_  dom  A
4 fdm 5755 . . . . . . . . 9  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
53, 4syl5sseq 3491 . . . . . . . 8  |-  ( A : NN --> NN0  ->  ( `' A " NN ) 
C_  NN )
6 simpl 463 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
75sselda 3443 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
86, 7ffvelrnd 6045 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
97nnnn0d 10953 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
108, 9nn0mulcld 10958 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1110nn0cnd 10955 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
12 simpr 467 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1312eldifad 3427 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1412eldifbd 3428 . . . . . . . . . . . . . 14  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
15 simpl 463 . . . . . . . . . . . . . . 15  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
16 ffn 5750 . . . . . . . . . . . . . . 15  |-  ( A : NN --> NN0  ->  A  Fn  NN )
17 elpreima 6024 . . . . . . . . . . . . . . 15  |-  ( A  Fn  NN  ->  (
k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
1815, 16, 173syl 18 . . . . . . . . . . . . . 14  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
1914, 18mtbid 306 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
20 imnan 428 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2119, 20sylibr 217 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( k  e.  NN  ->  -.  ( A `  k )  e.  NN ) )
2213, 21mpd 15 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( A `  k
)  e.  NN )
2315, 13ffvelrnd 6045 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
24 elnn0 10899 . . . . . . . . . . . 12  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2523, 24sylib 201 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
26 orel1 388 . . . . . . . . . . 11  |-  ( -.  ( A `  k
)  e.  NN  ->  ( ( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 )  ->  ( A `  k )  =  0 ) )
2722, 25, 26sylc 62 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  =  0 )
2827oveq1d 6329 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
2913nncnd 10652 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3029mul02d 9856 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( 0  x.  k
)  =  0 )
3128, 30eqtrd 2495 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
32 nnuz 11222 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
3332eqimssi 3497 . . . . . . . . 9  |-  NN  C_  ( ZZ>= `  1 )
3433a1i 11 . . . . . . . 8  |-  ( A : NN --> NN0  ->  NN  C_  ( ZZ>= `  1 )
)
355, 11, 31, 34sumss 13838 . . . . . . 7  |-  ( A : NN --> NN0  ->  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
)  =  sum_ k  e.  NN  ( ( A `
 k )  x.  k ) )
3635eqcomd 2467 . . . . . 6  |-  ( A : NN --> NN0  ->  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
3736adantr 471 . . . . 5  |-  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin )  ->  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
3837eqeq1d 2463 . . . 4  |-  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin )  -> 
( sum_ k  e.  NN  ( ( A `  k )  x.  k
)  =  N  <->  sum_ k  e.  ( `' A " NN ) ( ( A `
 k )  x.  k )  =  N ) )
3938pm5.32i 647 . . 3  |-  ( ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin )  /\  sum_ k  e.  NN  ( ( A `  k )  x.  k
)  =  N )  <-> 
( ( A : NN
--> NN0  /\  ( `' A " NN )  e.  Fin )  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
)  =  N ) )
40 df-3an 993 . . 3  |-  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N )  <->  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin )  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N ) )
41 df-3an 993 . . 3  |-  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k
)  x.  k )  =  N )  <->  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin )  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
)  =  N ) )
4239, 40, 413bitr4i 285 . 2  |-  ( ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N )  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k
)  x.  k )  =  N ) )
432, 42bitri 257 1  |-  ( A  e.  P  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  ( `' A " NN ) ( ( A `  k
)  x.  k )  =  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   {crab 2752    \ cdif 3412    C_ wss 3415   `'ccnv 4851   dom cdm 4852   "cima 4855    Fn wfn 5595   -->wf 5596   ` cfv 5600  (class class class)co 6314    ^m cmap 7497   Fincfn 7594   0cc0 9564   1c1 9565    x. cmul 9569   NNcn 10636   NN0cn0 10897   ZZ>=cuz 11187   sum_csu 13800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801
This theorem is referenced by: (None)
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