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Theorem eulerpartlemf 28506
Description: Lemma for eulerpart 28518: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-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
eulerpartlemf  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( A `  t
)  =  0 )
Distinct variable groups:    z, t    f, g, k, n, t, A    f, J    f, N    P, g
Allowed substitution hints:    A( x, y, z, r)    D( x, y, z, t, f, g, k, n, r)    P( x, y, z, t, f, k, n, r)    R( x, y, z, t, f, g, k, n, r)    T( x, y, z, t, f, g, k, n, r)    F( x, y, z, t, f, g, k, n, r)    H( x, y, z, t, f, g, k, n, r)    J( x, y, z, t, g, k, n, r)    M( x, y, z, t, f, g, k, n, r)    N( x, y, z, t, g, k, n, r)    O( x, y, z, t, f, g, k, n, r)

Proof of Theorem eulerpartlemf
StepHypRef Expression
1 eldif 3481 . . . . . 6  |-  ( t  e.  ( NN  \  J )  <->  ( t  e.  NN  /\  -.  t  e.  J ) )
2 breq2 4460 . . . . . . . . . . 11  |-  ( z  =  t  ->  (
2  ||  z  <->  2  ||  t ) )
32notbid 294 . . . . . . . . . 10  |-  ( z  =  t  ->  ( -.  2  ||  z  <->  -.  2  ||  t ) )
4 eulerpart.j . . . . . . . . . 10  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
53, 4elrab2 3259 . . . . . . . . 9  |-  ( t  e.  J  <->  ( t  e.  NN  /\  -.  2  ||  t ) )
65simplbi2 625 . . . . . . . 8  |-  ( t  e.  NN  ->  ( -.  2  ||  t  -> 
t  e.  J ) )
76con1d 124 . . . . . . 7  |-  ( t  e.  NN  ->  ( -.  t  e.  J  ->  2  ||  t ) )
87imp 429 . . . . . 6  |-  ( ( t  e.  NN  /\  -.  t  e.  J
)  ->  2  ||  t )
91, 8sylbi 195 . . . . 5  |-  ( t  e.  ( NN  \  J )  ->  2  ||  t )
109adantl 466 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
2  ||  t )
1110adantr 465 . . 3  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  2  ||  t )
12 simpll 753 . . . 4  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  A  e.  ( T  i^i  R
) )
13 eldifi 3622 . . . . . 6  |-  ( t  e.  ( NN  \  J )  ->  t  e.  NN )
14 eulerpart.p . . . . . . . . . . 11  |-  P  =  { f  e.  ( NN0  ^m  NN )  |  ( ( `' f " NN )  e.  Fin  /\  sum_ k  e.  NN  (
( f `  k
)  x.  k )  =  N ) }
15 eulerpart.o . . . . . . . . . . 11  |-  O  =  { g  e.  P  |  A. n  e.  ( `' g " NN )  -.  2  ||  n }
16 eulerpart.d . . . . . . . . . . 11  |-  D  =  { g  e.  P  |  A. n  e.  NN  ( g `  n
)  <_  1 }
17 eulerpart.f . . . . . . . . . . 11  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
18 eulerpart.h . . . . . . . . . . 11  |-  H  =  { r  e.  ( ( ~P NN0  i^i  Fin )  ^m  J )  |  ( r supp  (/) )  e. 
Fin }
19 eulerpart.m . . . . . . . . . . 11  |-  M  =  ( r  e.  H  |->  { <. x ,  y
>.  |  ( x  e.  J  /\  y  e.  ( r `  x
) ) } )
20 eulerpart.r . . . . . . . . . . 11  |-  R  =  { f  |  ( `' f " NN )  e.  Fin }
21 eulerpart.t . . . . . . . . . . 11  |-  T  =  { f  e.  ( NN0  ^m  NN )  |  ( `' f
" NN )  C_  J }
2214, 15, 16, 4, 17, 18, 19, 20, 21eulerpartlemt0 28505 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
2322simp1bi 1011 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
24 elmapi 7459 . . . . . . . . 9  |-  ( A  e.  ( NN0  ^m  NN )  ->  A : NN
--> NN0 )
2523, 24syl 16 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
26 ffn 5737 . . . . . . . 8  |-  ( A : NN --> NN0  ->  A  Fn  NN )
27 elpreima 6008 . . . . . . . 8  |-  ( A  Fn  NN  ->  (
t  e.  ( `' A " NN )  <-> 
( t  e.  NN  /\  ( A `  t
)  e.  NN ) ) )
2825, 26, 273syl 20 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( t  e.  ( `' A " NN )  <->  ( t  e.  NN  /\  ( A `
 t )  e.  NN ) ) )
2928baibd 909 . . . . . 6  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  NN )  ->  ( t  e.  ( `' A " NN )  <-> 
( A `  t
)  e.  NN ) )
3013, 29sylan2 474 . . . . 5  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( t  e.  ( `' A " NN )  <-> 
( A `  t
)  e.  NN ) )
3130biimpar 485 . . . 4  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  t  e.  ( `' A " NN ) )
3222simp3bi 1013 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  C_  J )
3332sselda 3499 . . . . 5  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( `' A " NN ) )  ->  t  e.  J
)
345simprbi 464 . . . . 5  |-  ( t  e.  J  ->  -.  2  ||  t )
3533, 34syl 16 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( `' A " NN ) )  ->  -.  2  ||  t )
3612, 31, 35syl2anc 661 . . 3  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  -.  2  ||  t )
3711, 36pm2.65da 576 . 2  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  ->  -.  ( A `  t
)  e.  NN )
3825adantr 465 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  ->  A : NN --> NN0 )
3913adantl 466 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
t  e.  NN )
4038, 39ffvelrnd 6033 . . 3  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( A `  t
)  e.  NN0 )
41 elnn0 10818 . . 3  |-  ( ( A `  t )  e.  NN0  <->  ( ( A `
 t )  e.  NN  \/  ( A `
 t )  =  0 ) )
4240, 41sylib 196 . 2  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( ( A `  t )  e.  NN  \/  ( A `  t
)  =  0 ) )
43 orel1 382 . 2  |-  ( -.  ( A `  t
)  e.  NN  ->  ( ( ( A `  t )  e.  NN  \/  ( A `  t
)  =  0 )  ->  ( A `  t )  =  0 ) )
4437, 42, 43sylc 60 1  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( A `  t
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   class class class wbr 4456   {copab 4514    |-> cmpt 4515   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   supp csupp 6917    ^m cmap 7438   Fincfn 7535   0cc0 9509   1c1 9510    x. cmul 9514    <_ cle 9646   NNcn 10556   2c2 10606   NN0cn0 10816   ^cexp 12169   sum_csu 13520    || cdvds 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-mulcl 9571  ax-i2m1 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-n0 10817
This theorem is referenced by:  eulerpartlemgh  28514
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