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Theorem eulerpartlemelr 29016
Description: Lemma for eulerpart 29041. (Contributed by Thierry Arnoux, 8-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
eulerpartlemelr  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
Distinct variable groups:    f, k, A    R, f, k
Allowed substitution hints:    S( f, k)

Proof of Theorem eulerpartlemelr
StepHypRef Expression
1 inss1 3688 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  ( NN0  ^m  NN )
21sseli 3466 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  ( NN0  ^m  NN ) )
3 elmapi 7501 . . 3  |-  ( A  e.  ( NN0  ^m  NN )  ->  A : NN
--> NN0 )
42, 3syl 17 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
5 inss2 3689 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  R
65sseli 3466 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  R )
7 cnveq 5028 . . . . . 6  |-  ( f  =  A  ->  `' f  =  `' A
)
87imaeq1d 5187 . . . . 5  |-  ( f  =  A  ->  ( `' f " NN )  =  ( `' A " NN ) )
98eleq1d 2498 . . . 4  |-  ( f  =  A  ->  (
( `' f " NN )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
10 eulerpartlems.r . . . 4  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
119, 10elab2g 3226 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A  e.  R  <->  ( `' A " NN )  e. 
Fin ) )
126, 11mpbid 213 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( `' A " NN )  e.  Fin )
134, 12jca 534 1  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A : NN --> NN0  /\  ( `' A " NN )  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414    i^i cin 3441    |-> cmpt 4484   `'ccnv 4853   "cima 4857   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   Fincfn 7577    x. cmul 9543   NNcn 10609   NN0cn0 10869   sum_csu 13730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482
This theorem is referenced by:  eulerpartlemsv2  29017  eulerpartlemsf  29018  eulerpartlems  29019  eulerpartlemsv3  29020  eulerpartlemgc  29021
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