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

Theorem ballotlemgun 24774
Description: A property of the defined  .^ operator (Contributed by Thierry Arnoux, 26-Apr-2017.)
Hypotheses
Ref Expression
ballotth.m  |-  M  e.  NN
ballotth.n  |-  N  e.  NN
ballotth.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
ballotth.p  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
ballotth.f  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
ballotth.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotth.mgtn  |-  N  < 
M
ballotth.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
ballotth.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotth.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
ballotlemg  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
ballotlemgun.1  |-  ( ph  ->  U  e.  Fin )
ballotlemgun.2  |-  ( ph  ->  V  e.  Fin )
ballotlemgun.3  |-  ( ph  ->  W  e.  Fin )
ballotlemgun.4  |-  ( ph  ->  ( V  i^i  W
)  =  (/) )
Assertion
Ref Expression
ballotlemgun  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O    k, M    k, N    k, O    i, c, F, k   
i, E, k    k, I, c    E, c    i, I, c    S, k, i, c    R, i    v, u, I    u, R, v   
u, S, v    u, U, v    u, V, v   
u, W, v
Allowed substitution hints:    ph( x, v, u, i, k, c)    P( x, v, u, i, k, c)    R( x, k, c)    S( x)    U( x, i, k, c)    E( x, v, u)    .^ ( x, v, u, i, k, c)    F( x, v, u)    I( x)    M( x, v, u)    N( x, v, u)    O( x, v, u)    V( x, i, k, c)    W( x, i, k, c)

Proof of Theorem ballotlemgun
StepHypRef Expression
1 indir 3581 . . . . . 6  |-  ( ( V  u.  W )  i^i  U )  =  ( ( V  i^i  U )  u.  ( W  i^i  U ) )
21fveq2i 5723 . . . . 5  |-  ( # `  ( ( V  u.  W )  i^i  U
) )  =  (
# `  ( ( V  i^i  U )  u.  ( W  i^i  U
) ) )
3 ballotlemgun.2 . . . . . . 7  |-  ( ph  ->  V  e.  Fin )
4 infi 7324 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  i^i  U )  e. 
Fin )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  ( V  i^i  U
)  e.  Fin )
6 ballotlemgun.3 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
7 infi 7324 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  i^i  U )  e. 
Fin )
86, 7syl 16 . . . . . 6  |-  ( ph  ->  ( W  i^i  U
)  e.  Fin )
9 ballotlemgun.4 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  W
)  =  (/) )
109ineq1d 3533 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  i^i  U )  =  ( (/)  i^i  U
) )
11 inindir 3551 . . . . . . 7  |-  ( ( V  i^i  W )  i^i  U )  =  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )
12 incom 3525 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (
(/)  i^i  U )
13 in0 3645 . . . . . . . 8  |-  ( U  i^i  (/) )  =  (/)
1412, 13eqtr3i 2457 . . . . . . 7  |-  ( (/)  i^i 
U )  =  (/)
1510, 11, 143eqtr3g 2490 . . . . . 6  |-  ( ph  ->  ( ( V  i^i  U )  i^i  ( W  i^i  U ) )  =  (/) )
16 hashun 11648 . . . . . 6  |-  ( ( ( V  i^i  U
)  e.  Fin  /\  ( W  i^i  U )  e.  Fin  /\  (
( V  i^i  U
)  i^i  ( W  i^i  U ) )  =  (/) )  ->  ( # `  ( ( V  i^i  U )  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
175, 8, 15, 16syl3anc 1184 . . . . 5  |-  ( ph  ->  ( # `  (
( V  i^i  U
)  u.  ( W  i^i  U ) ) )  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
182, 17syl5eq 2479 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  i^i  U )
)  =  ( (
# `  ( V  i^i  U ) )  +  ( # `  ( W  i^i  U ) ) ) )
19 difundir 3586 . . . . . 6  |-  ( ( V  u.  W ) 
\  U )  =  ( ( V  \  U )  u.  ( W  \  U ) )
2019fveq2i 5723 . . . . 5  |-  ( # `  ( ( V  u.  W )  \  U
) )  =  (
# `  ( ( V  \  U )  u.  ( W  \  U
) ) )
21 diffi 7331 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  U )  e. 
Fin )
223, 21syl 16 . . . . . 6  |-  ( ph  ->  ( V  \  U
)  e.  Fin )
23 diffi 7331 . . . . . . 7  |-  ( W  e.  Fin  ->  ( W  \  U )  e. 
Fin )
246, 23syl 16 . . . . . 6  |-  ( ph  ->  ( W  \  U
)  e.  Fin )
259difeq1d 3456 . . . . . . 7  |-  ( ph  ->  ( ( V  i^i  W )  \  U )  =  ( (/)  \  U
) )
26 difindir 3588 . . . . . . 7  |-  ( ( V  i^i  W ) 
\  U )  =  ( ( V  \  U )  i^i  ( W  \  U ) )
27 0dif 3691 . . . . . . 7  |-  ( (/)  \  U )  =  (/)
2825, 26, 273eqtr3g 2490 . . . . . 6  |-  ( ph  ->  ( ( V  \  U )  i^i  ( W  \  U ) )  =  (/) )
29 hashun 11648 . . . . . 6  |-  ( ( ( V  \  U
)  e.  Fin  /\  ( W  \  U )  e.  Fin  /\  (
( V  \  U
)  i^i  ( W  \  U ) )  =  (/) )  ->  ( # `  ( ( V  \  U )  u.  ( W  \  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3022, 24, 28, 29syl3anc 1184 . . . . 5  |-  ( ph  ->  ( # `  (
( V  \  U
)  u.  ( W 
\  U ) ) )  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3120, 30syl5eq 2479 . . . 4  |-  ( ph  ->  ( # `  (
( V  u.  W
)  \  U )
)  =  ( (
# `  ( V  \  U ) )  +  ( # `  ( W  \  U ) ) ) )
3218, 31oveq12d 6091 . . 3  |-  ( ph  ->  ( ( # `  (
( V  u.  W
)  i^i  U )
)  -  ( # `  ( ( V  u.  W )  \  U
) ) )  =  ( ( ( # `  ( V  i^i  U
) )  +  (
# `  ( W  i^i  U ) ) )  -  ( ( # `  ( V  \  U
) )  +  (
# `  ( W  \  U ) ) ) ) )
33 hashcl 11631 . . . . . 6  |-  ( ( V  i^i  U )  e.  Fin  ->  ( # `
 ( V  i^i  U ) )  e.  NN0 )
343, 4, 333syl 19 . . . . 5  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  NN0 )
3534nn0cnd 10268 . . . 4  |-  ( ph  ->  ( # `  ( V  i^i  U ) )  e.  CC )
36 hashcl 11631 . . . . . 6  |-  ( ( W  i^i  U )  e.  Fin  ->  ( # `
 ( W  i^i  U ) )  e.  NN0 )
376, 7, 363syl 19 . . . . 5  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  NN0 )
3837nn0cnd 10268 . . . 4  |-  ( ph  ->  ( # `  ( W  i^i  U ) )  e.  CC )
39 hashcl 11631 . . . . . 6  |-  ( ( V  \  U )  e.  Fin  ->  ( # `
 ( V  \  U ) )  e. 
NN0 )
403, 21, 393syl 19 . . . . 5  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  NN0 )
4140nn0cnd 10268 . . . 4  |-  ( ph  ->  ( # `  ( V  \  U ) )  e.  CC )
42 hashcl 11631 . . . . . 6  |-  ( ( W  \  U )  e.  Fin  ->  ( # `
 ( W  \  U ) )  e. 
NN0 )
436, 23, 423syl 19 . . . . 5  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  NN0 )
4443nn0cnd 10268 . . . 4  |-  ( ph  ->  ( # `  ( W  \  U ) )  e.  CC )
4535, 38, 41, 44addsub4d 9450 . . 3  |-  ( ph  ->  ( ( ( # `  ( V  i^i  U
) )  +  (
# `  ( W  i^i  U ) ) )  -  ( ( # `  ( V  \  U
) )  +  (
# `  ( W  \  U ) ) ) )  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
4632, 45eqtrd 2467 . 2  |-  ( ph  ->  ( ( # `  (
( V  u.  W
)  i^i  U )
)  -  ( # `  ( ( V  u.  W )  \  U
) ) )  =  ( ( ( # `  ( V  i^i  U
) )  -  ( # `
 ( V  \  U ) ) )  +  ( ( # `  ( W  i^i  U
) )  -  ( # `
 ( W  \  U ) ) ) ) )
47 ballotlemgun.1 . . 3  |-  ( ph  ->  U  e.  Fin )
48 unfi 7366 . . . 4  |-  ( ( V  e.  Fin  /\  W  e.  Fin )  ->  ( V  u.  W
)  e.  Fin )
493, 6, 48syl2anc 643 . . 3  |-  ( ph  ->  ( V  u.  W
)  e.  Fin )
50 ballotth.m . . . 4  |-  M  e.  NN
51 ballotth.n . . . 4  |-  N  e.  NN
52 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
53 ballotth.p . . . 4  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
54 ballotth.f . . . 4  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
55 ballotth.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
56 ballotth.mgtn . . . 4  |-  N  < 
M
57 ballotth.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
58 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
59 ballotth.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
60 ballotlemg . . . 4  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
6150, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 24773 . . 3  |-  ( ( U  e.  Fin  /\  ( V  u.  W
)  e.  Fin )  ->  ( U  .^  ( V  u.  W )
)  =  ( (
# `  ( ( V  u.  W )  i^i  U ) )  -  ( # `  ( ( V  u.  W ) 
\  U ) ) ) )
6247, 49, 61syl2anc 643 . 2  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( (
# `  ( ( V  u.  W )  i^i  U ) )  -  ( # `  ( ( V  u.  W ) 
\  U ) ) ) )
6350, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 24773 . . . 4  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6447, 3, 63syl2anc 643 . . 3  |-  ( ph  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
6550, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60ballotlemgval 24773 . . . 4  |-  ( ( U  e.  Fin  /\  W  e.  Fin )  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6647, 6, 65syl2anc 643 . . 3  |-  ( ph  ->  ( U  .^  W
)  =  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) )
6764, 66oveq12d 6091 . 2  |-  ( ph  ->  ( ( U  .^  V )  +  ( U  .^  W )
)  =  ( ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) )  +  ( (
# `  ( W  i^i  U ) )  -  ( # `  ( W 
\  U ) ) ) ) )
6846, 62, 673eqtr4d 2477 1  |-  ( ph  ->  ( U  .^  ( V  u.  W )
)  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   ifcif 3731   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Fincfn 7101   supcsup 7437   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ...cfz 11035   #chash 11610
This theorem is referenced by:  ballotlemfrceq  24778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-hash 11611
  Copyright terms: Public domain W3C validator