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Theorem eulerpartlemf 29203
Description: Lemma for eulerpart 29215: 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 3414 . . . . . 6  |-  ( t  e.  ( NN  \  J )  <->  ( t  e.  NN  /\  -.  t  e.  J ) )
2 breq2 4406 . . . . . . . . . . 11  |-  ( z  =  t  ->  (
2  ||  z  <->  2  ||  t ) )
32notbid 296 . . . . . . . . . 10  |-  ( z  =  t  ->  ( -.  2  ||  z  <->  -.  2  ||  t ) )
4 eulerpart.j . . . . . . . . . 10  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
53, 4elrab2 3198 . . . . . . . . 9  |-  ( t  e.  J  <->  ( t  e.  NN  /\  -.  2  ||  t ) )
65simplbi2 631 . . . . . . . 8  |-  ( t  e.  NN  ->  ( -.  2  ||  t  -> 
t  e.  J ) )
76con1d 128 . . . . . . 7  |-  ( t  e.  NN  ->  ( -.  t  e.  J  ->  2  ||  t ) )
87imp 431 . . . . . 6  |-  ( ( t  e.  NN  /\  -.  t  e.  J
)  ->  2  ||  t )
91, 8sylbi 199 . . . . 5  |-  ( t  e.  ( NN  \  J )  ->  2  ||  t )
109adantl 468 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
2  ||  t )
1110adantr 467 . . 3  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  2  ||  t )
12 simpll 760 . . . 4  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  A  e.  ( T  i^i  R
) )
13 eldifi 3555 . . . . . 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 29202 . . . . . . . . . 10  |-  ( A  e.  ( T  i^i  R )  <->  ( A  e.  ( NN0  ^m  NN )  /\  ( `' A " NN )  e.  Fin  /\  ( `' A " NN )  C_  J ) )
2322simp1bi 1023 . . . . . . . . 9  |-  ( A  e.  ( T  i^i  R )  ->  A  e.  ( NN0  ^m  NN ) )
24 elmapi 7493 . . . . . . . . 9  |-  ( A  e.  ( NN0  ^m  NN )  ->  A : NN
--> NN0 )
2523, 24syl 17 . . . . . . . 8  |-  ( A  e.  ( T  i^i  R )  ->  A : NN
--> NN0 )
26 ffn 5728 . . . . . . . 8  |-  ( A : NN --> NN0  ->  A  Fn  NN )
27 elpreima 6002 . . . . . . . 8  |-  ( A  Fn  NN  ->  (
t  e.  ( `' A " NN )  <-> 
( t  e.  NN  /\  ( A `  t
)  e.  NN ) ) )
2825, 26, 273syl 18 . . . . . . 7  |-  ( A  e.  ( T  i^i  R )  ->  ( t  e.  ( `' A " NN )  <->  ( t  e.  NN  /\  ( A `
 t )  e.  NN ) ) )
2928baibd 920 . . . . . 6  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  NN )  ->  ( t  e.  ( `' A " NN )  <-> 
( A `  t
)  e.  NN ) )
3013, 29sylan2 477 . . . . 5  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( t  e.  ( `' A " NN )  <-> 
( A `  t
)  e.  NN ) )
3130biimpar 488 . . . 4  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  t  e.  ( `' A " NN ) )
3222simp3bi 1025 . . . . . 6  |-  ( A  e.  ( T  i^i  R )  ->  ( `' A " NN )  C_  J )
3332sselda 3432 . . . . 5  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( `' A " NN ) )  ->  t  e.  J
)
345simprbi 466 . . . . 5  |-  ( t  e.  J  ->  -.  2  ||  t )
3533, 34syl 17 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( `' A " NN ) )  ->  -.  2  ||  t )
3612, 31, 35syl2anc 667 . . 3  |-  ( ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  /\  ( A `
 t )  e.  NN )  ->  -.  2  ||  t )
3711, 36pm2.65da 580 . 2  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  ->  -.  ( A `  t
)  e.  NN )
3825adantr 467 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  ->  A : NN --> NN0 )
3913adantl 468 . . . 4  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
t  e.  NN )
4038, 39ffvelrnd 6023 . . 3  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( A `  t
)  e.  NN0 )
41 elnn0 10871 . . 3  |-  ( ( A `  t )  e.  NN0  <->  ( ( A `
 t )  e.  NN  \/  ( A `
 t )  =  0 ) )
4240, 41sylib 200 . 2  |-  ( ( A  e.  ( T  i^i  R )  /\  t  e.  ( NN  \  J ) )  -> 
( ( A `  t )  e.  NN  \/  ( A `  t
)  =  0 ) )
43 orel1 384 . 2  |-  ( -.  ( A `  t
)  e.  NN  ->  ( ( ( A `  t )  e.  NN  \/  ( A `  t
)  =  0 )  ->  ( A `  t )  =  0 ) )
4437, 42, 43sylc 62 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 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   {crab 2741    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   class class class wbr 4402   {copab 4460    |-> cmpt 4461   `'ccnv 4833   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   supp csupp 6914    ^m cmap 7472   Fincfn 7569   0cc0 9539   1c1 9540    x. cmul 9544    <_ cle 9676   NNcn 10609   2c2 10659   NN0cn0 10869   ^cexp 12272   sum_csu 13752    || cdvds 14305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-map 7474  df-n0 10870
This theorem is referenced by:  eulerpartlemgh  29211
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