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Theorem eulerpartlemv 29149
Description: Lemma for eulerpart 29167. (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 29148 . 2  |-  ( A  e.  P  <->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  N ) )
3 cnvimass 5150 . . . . . . . . 9  |-  ( `' A " NN ) 
C_  dom  A
4 fdm 5693 . . . . . . . . 9  |-  ( A : NN --> NN0  ->  dom 
A  =  NN )
53, 4syl5sseq 3455 . . . . . . . 8  |-  ( A : NN --> NN0  ->  ( `' A " NN ) 
C_  NN )
6 simpl 458 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  A : NN --> NN0 )
75sselda 3407 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN )
86, 7ffvelrnd 5982 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( A `  k )  e.  NN0 )
97nnnn0d 10876 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  k  e.  NN0 )
108, 9nn0mulcld 10881 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  NN0 )
1110nn0cnd 10878 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( `' A " NN ) )  ->  ( ( A `
 k )  x.  k )  e.  CC )
12 simpr 462 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  ( NN 
\  ( `' A " NN ) ) )
1312eldifad 3391 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  NN )
1412eldifbd 3392 . . . . . . . . . . . . . 14  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  k  e.  ( `' A " NN ) )
15 simpl 458 . . . . . . . . . . . . . . 15  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  A : NN --> NN0 )
16 ffn 5689 . . . . . . . . . . . . . . 15  |-  ( A : NN --> NN0  ->  A  Fn  NN )
17 elpreima 5961 . . . . . . . . . . . . . . 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 301 . . . . . . . . . . . . 13  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  ->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
20 imnan 423 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN  ->  -.  ( A `  k
)  e.  NN )  <->  -.  ( k  e.  NN  /\  ( A `  k
)  e.  NN ) )
2119, 20sylibr 215 . . . . . . . . . . . 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 5982 . . . . . . . . . . . 12  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( A `  k
)  e.  NN0 )
24 elnn0 10822 . . . . . . . . . . . 12  |-  ( ( A `  k )  e.  NN0  <->  ( ( A `
 k )  e.  NN  \/  ( A `
 k )  =  0 ) )
2523, 24sylib 199 . . . . . . . . . . 11  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  e.  NN  \/  ( A `  k
)  =  0 ) )
26 orel1 383 . . . . . . . . . . 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 6264 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  ( 0  x.  k ) )
2913nncnd 10576 . . . . . . . . . 10  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
k  e.  CC )
3029mul02d 9782 . . . . . . . . 9  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( 0  x.  k
)  =  0 )
3128, 30eqtrd 2462 . . . . . . . 8  |-  ( ( A : NN --> NN0  /\  k  e.  ( NN  \  ( `' A " NN ) ) )  -> 
( ( A `  k )  x.  k
)  =  0 )
32 nnuz 11145 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
3332eqimssi 3461 . . . . . . . . 9  |-  NN  C_  ( ZZ>= `  1 )
3433a1i 11 . . . . . . . 8  |-  ( A : NN --> NN0  ->  NN  C_  ( ZZ>= `  1 )
)
355, 11, 31, 34sumss 13733 . . . . . . 7  |-  ( A : NN --> NN0  ->  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
)  =  sum_ k  e.  NN  ( ( A `
 k )  x.  k ) )
3635eqcomd 2434 . . . . . 6  |-  ( A : NN --> NN0  ->  sum_ k  e.  NN  (
( A `  k
)  x.  k )  =  sum_ k  e.  ( `' A " NN ) ( ( A `  k )  x.  k
) )
3736adantr 466 . . . . 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 2430 . . . 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 641 . . 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 984 . . 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 984 . . 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 280 . 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 252 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {crab 2718    \ cdif 3376    C_ wss 3379   `'ccnv 4795   dom cdm 4796   "cima 4799    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249    ^m cmap 7427   Fincfn 7524   0cc0 9490   1c1 9491    x. cmul 9495   NNcn 10560   NN0cn0 10820   ZZ>=cuz 11110   sum_csu 13695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-oi 7978  df-card 8325  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fz 11736  df-fzo 11867  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-sum 13696
This theorem is referenced by: (None)
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