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Theorem eulerpartlemelr 26762
Description: Lemma for eulerpart 26787. (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 3591 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  ( NN0  ^m  NN )
21sseli 3373 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  ( NN0  ^m  NN ) )
3 elmapi 7255 . . 3  |-  ( A  e.  ( NN0  ^m  NN )  ->  A : NN
--> NN0 )
42, 3syl 16 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A : NN --> NN0 )
5 inss2 3592 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  R
65sseli 3373 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  R )
7 cnveq 5034 . . . . . 6  |-  ( f  =  A  ->  `' f  =  `' A
)
87imaeq1d 5189 . . . . 5  |-  ( f  =  A  ->  ( `' f " NN )  =  ( `' A " NN ) )
98eleq1d 2509 . . . 4  |-  ( f  =  A  ->  (
( `' f " NN )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
10 eulerpartlems.r . . . 4  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
119, 10elab2g 3129 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( A  e.  R  <->  ( `' A " NN )  e. 
Fin ) )
126, 11mpbid 210 . 2  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  ( `' A " NN )  e.  Fin )
134, 12jca 532 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 369    = wceq 1369    e. wcel 1756   {cab 2429    i^i cin 3348    e. cmpt 4371   `'ccnv 4860   "cima 4864   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^m cmap 7235   Fincfn 7331    x. cmul 9308   NNcn 10343   NN0cn0 10600   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-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  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-ral 2741  df-rex 2742  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-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  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-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-map 7237
This theorem is referenced by:  eulerpartlemsv2  26763  eulerpartlemsf  26764  eulerpartlems  26765  eulerpartlemsv3  26766  eulerpartlemgc  26767
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