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Theorem eulerpartlemt0 28572
Description: Lemma for eulerpart 28585 (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
eulerpart.p  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
eulerpart.o  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
eulerpart.d  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
eulerpart.j  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
eulerpart.f  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
eulerpart.h  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
eulerpart.m  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
eulerpart.r  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
eulerpart.t  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
Assertion
Ref Expression
eulerpartlemt0  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
Distinct variable groups:    A, f    f, J
Allowed substitution hints:    A( x, y, z, g, k, n, r)    D( x, y, z, f, g, k, n, r)    P( x, y, z, f, g, k, n, r)    R( x, y, z, f, g, k, n, r)    T( x, y, z, f, g, k, n, r)    F( x, y, z, f, g, k, n, r)    H( x, y, z, f, g, k, n, r)    J( x, y, z, g, k, n, r)    M( x, y, z, f, g, k, n, r)    N( x, y, z, f, g, k, n, r)    O( x, y, z, f, g, k, n, r)

Proof of Theorem eulerpartlemt0
StepHypRef Expression
1 cnveq 5165 . . . . . 6  |-  ( f  =  A  ->  `' f  =  `' A
)
21imaeq1d 5324 . . . . 5  |-  ( f  =  A  ->  ( `' f " NN )  =  ( `' A " NN ) )
32sseq1d 3516 . . . 4  |-  ( f  =  A  ->  (
( `' f " NN )  C_  J  <->  ( `' A " NN )  C_  J ) )
4 eulerpart.t . . . 4  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
53, 4elrab2 3256 . . 3  |-  ( A  e.  T  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  C_  J
) )
62eleq1d 2523 . . . 4  |-  ( f  =  A  ->  (
( `' f " NN )  e.  Fin  <->  ( `' A " NN )  e.  Fin ) )
7 eulerpart.r . . . 4  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
86, 7elab4g 3247 . . 3  |-  ( A  e.  R  <->  ( A  e.  _V  /\  ( `' A " NN )  e.  Fin ) )
95, 8anbi12i 695 . 2  |-  ( ( A  e.  T  /\  A  e.  R )  <->  ( ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  C_  J
)  /\  ( A  e.  _V  /\  ( `' A " NN )  e.  Fin ) ) )
10 elin 3673 . 2  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  T  /\  A  e.  R ) )
11 elex 3115 . . . . 5  |-  ( A  e.  ( NN0  ^m  NN )  ->  A  e. 
_V )
1211pm4.71i 630 . . . 4  |-  ( A  e.  ( NN0  ^m  NN )  <->  ( A  e.  ( NN0  ^m  NN )  /\  A  e.  _V ) )
1312anbi1i 693 . . 3  |-  ( ( A  e.  ( NN0 
^m  NN )  /\  ( ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )  <->  ( ( A  e.  ( NN0  ^m  NN )  /\  A  e. 
_V )  /\  (
( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) ) )
14 3anass 975 . . 3  |-  ( ( A  e.  ( NN0 
^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN ) 
C_  J )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( ( `' A " NN )  e.  Fin  /\  ( `' A " NN ) 
C_  J ) ) )
15 an42 823 . . 3  |-  ( ( ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  C_  J
)  /\  ( A  e.  _V  /\  ( `' A " NN )  e.  Fin ) )  <-> 
( ( A  e.  ( NN0  ^m  NN )  /\  A  e.  _V )  /\  ( ( `' A " NN )  e.  Fin  /\  ( `' A " NN ) 
C_  J ) ) )
1613, 14, 153bitr4i 277 . 2  |-  ( ( A  e.  ( NN0 
^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN ) 
C_  J )  <->  ( ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN ) 
C_  J )  /\  ( A  e.  _V  /\  ( `' A " NN )  e.  Fin ) ) )
179, 10, 163bitr4i 277 1  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   class class class wbr 4439   {copab 4496    |-> cmpt 4497   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   supp csupp 6891    ^m cmap 7412   Fincfn 7509   1c1 9482    x. cmul 9486    <_ cle 9618   NNcn 10531   2c2 10581   NN0cn0 10791   ^cexp 12148   sum_csu 13590    || cdvds 14070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  eulerpartlemf  28573  eulerpartlemt  28574  eulerpartlemmf  28578  eulerpartlemgvv  28579  eulerpartlemgu  28580  eulerpartlemgh  28581  eulerpartlemgs2  28583  eulerpartlemn  28584
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