Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemelr Structured version   Unicode version

Theorem eulerpartlemelr 27964
Description: Lemma for eulerpart 27989. (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 3718 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  ( NN0  ^m  NN )
21sseli 3500 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  ( NN0  ^m  NN ) )
3 elmapi 7440 . . 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 3719 . . . 4  |-  ( ( NN0  ^m  NN )  i^i  R )  C_  R
65sseli 3500 . . 3  |-  ( A  e.  ( ( NN0 
^m  NN )  i^i 
R )  ->  A  e.  R )
7 cnveq 5176 . . . . . 6  |-  ( f  =  A  ->  `' f  =  `' A
)
87imaeq1d 5336 . . . . 5  |-  ( f  =  A  ->  ( `' f " NN )  =  ( `' A " NN ) )
98eleq1d 2536 . . . 4  |-  ( f  =  A  ->  (
( `' f " NN )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
10 eulerpartlems.r . . . 4  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
119, 10elab2g 3252 . . 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 1379    e. wcel 1767   {cab 2452    i^i cin 3475    |-> cmpt 4505   `'ccnv 4998   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   Fincfn 7516    x. cmul 9497   NNcn 10536   NN0cn0 10795   sum_csu 13471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422
This theorem is referenced by:  eulerpartlemsv2  27965  eulerpartlemsf  27966  eulerpartlems  27967  eulerpartlemsv3  27968  eulerpartlemgc  27969
  Copyright terms: Public domain W3C validator