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Theorem eulerpartlemv 26769
Description: Lemma for eulerpart 26787 (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 26768 . 2  |-  ( A  e.  P  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N ) )
3 cnvimass 5210 . . . . . . . . 9  |-  ( `' A " NN ) 
C_  dom  A
4 fdm 5584 . . . . . . . . 9  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
53, 4syl5sseq 3425 . . . . . . . 8  |-  ( A : NN --> NN0  ->  ( `' A " NN ) 
C_  NN )
6 simpl 457 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
75sselda 3377 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
86, 7ffvelrnd 5865 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
97nnnn0d 10657 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
108, 9nn0mulcld 10662 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1110nn0cnd 10659 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
12 simpr 461 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1312eldifad 3361 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1412eldifbd 3362 . . . . . . . . . . . . . 14  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
15 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
16 ffn 5580 . . . . . . . . . . . . . . 15  |-  ( A : NN --> NN0  ->  A  Fn  NN )
17 elpreima 5844 . . . . . . . . . . . . . . 15  |-  ( A  Fn  NN  ->  (
k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
1815, 16, 173syl 20 . . . . . . . . . . . . . 14  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( k  e.  ( `' A " NN )  <-> 
( k  e.  NN  /\  ( A `  k
)  e.  NN ) ) )
1914, 18mtbid 300 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
20 imnan 422 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2119, 20sylibr 212 . . . . . . . . . . . 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 5865 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
24 elnn0 10602 . . . . . . . . . . . 12  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2523, 24sylib 196 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
26 orel1 382 . . . . . . . . . . 11  |-  ( -.  ( A `  k
)  e.  NN  ->  ( ( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 )  ->  ( A `  k )  =  0 ) )
2722, 25, 26sylc 60 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  =  0 )
2827oveq1d 6127 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
2913nncnd 10359 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3029mul02d 9588 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( 0  x.  k
)  =  0 )
3128, 30eqtrd 2475 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
32 nnuz 10917 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
3332eqimssi 3431 . . . . . . . . 9  |-  NN  C_  ( ZZ>= `  1 )
3433a1i 11 . . . . . . . 8  |-  ( A : NN --> NN0  ->  NN  C_  ( ZZ>= `  1 )
)
355, 11, 31, 34sumss 13222 . . . . . . 7  |-  ( A : NN --> NN0  ->  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
)  =  sum_ k  e.  NN  ( ( A `
 k )  x.  k ) )
3635eqcomd 2448 . . . . . 6  |-  ( A : NN --> NN0  ->  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
3736adantr 465 . . . . 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 2451 . . . 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 637 . . 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 967 . . 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 967 . . 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 277 . 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 249 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2740    \ cdif 3346    C_ wss 3349   `'ccnv 4860   dom cdm 4861   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^m cmap 7235   Fincfn 7331   0cc0 9303   1c1 9304    x. cmul 9308   NNcn 10343   NN0cn0 10600   ZZ>=cuz 10882   sum_csu 13184
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185
This theorem is referenced by: (None)
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