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Theorem ballotlemgval 23082
Description: Expand the value of  .^. (Contributed by Thierry Arnoux, 21-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
) ) ) )
Assertion
Ref Expression
ballotlemgval  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
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
Allowed substitution hints:    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)

Proof of Theorem ballotlemgval
StepHypRef Expression
1 ineq2 3364 . . . 4  |-  ( u  =  U  ->  (
v  i^i  u )  =  ( v  i^i 
U ) )
21fveq2d 5529 . . 3  |-  ( u  =  U  ->  ( # `
 ( v  i^i  u ) )  =  ( # `  (
v  i^i  U )
) )
3 difeq2 3288 . . . 4  |-  ( u  =  U  ->  (
v  \  u )  =  ( v  \  U ) )
43fveq2d 5529 . . 3  |-  ( u  =  U  ->  ( # `
 ( v  \  u ) )  =  ( # `  (
v  \  U )
) )
52, 4oveq12d 5876 . 2  |-  ( u  =  U  ->  (
( # `  ( v  i^i  u ) )  -  ( # `  (
v  \  u )
) )  =  ( ( # `  (
v  i^i  U )
)  -  ( # `  ( v  \  U
) ) ) )
6 ineq1 3363 . . . 4  |-  ( v  =  V  ->  (
v  i^i  U )  =  ( V  i^i  U ) )
76fveq2d 5529 . . 3  |-  ( v  =  V  ->  ( # `
 ( v  i^i 
U ) )  =  ( # `  ( V  i^i  U ) ) )
8 difeq1 3287 . . . 4  |-  ( v  =  V  ->  (
v  \  U )  =  ( V  \  U ) )
98fveq2d 5529 . . 3  |-  ( v  =  V  ->  ( # `
 ( v  \  U ) )  =  ( # `  ( V  \  U ) ) )
107, 9oveq12d 5876 . 2  |-  ( v  =  V  ->  (
( # `  ( v  i^i  U ) )  -  ( # `  (
v  \  U )
) )  =  ( ( # `  ( V  i^i  U ) )  -  ( # `  ( V  \  U ) ) ) )
11 ballotlemg . 2  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
12 ovex 5883 . 2  |-  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) )  e.  _V
135, 10, 11, 12ovmpt2 5983 1  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Fincfn 6863   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemgun  23083  ballotlemfg  23084  ballotlemfrc  23085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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