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Theorem ballotlemgval 28954
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 3634 . . . 4  |-  ( u  =  U  ->  (
v  i^i  u )  =  ( v  i^i 
U ) )
21fveq2d 5852 . . 3  |-  ( u  =  U  ->  ( # `
 ( v  i^i  u ) )  =  ( # `  (
v  i^i  U )
) )
3 difeq2 3554 . . . 4  |-  ( u  =  U  ->  (
v  \  u )  =  ( v  \  U ) )
43fveq2d 5852 . . 3  |-  ( u  =  U  ->  ( # `
 ( v  \  u ) )  =  ( # `  (
v  \  U )
) )
52, 4oveq12d 6295 . 2  |-  ( u  =  U  ->  (
( # `  ( v  i^i  u ) )  -  ( # `  (
v  \  u )
) )  =  ( ( # `  (
v  i^i  U )
)  -  ( # `  ( v  \  U
) ) ) )
6 ineq1 3633 . . . 4  |-  ( v  =  V  ->  (
v  i^i  U )  =  ( V  i^i  U ) )
76fveq2d 5852 . . 3  |-  ( v  =  V  ->  ( # `
 ( v  i^i 
U ) )  =  ( # `  ( V  i^i  U ) ) )
8 difeq1 3553 . . . 4  |-  ( v  =  V  ->  (
v  \  U )  =  ( V  \  U ) )
98fveq2d 5852 . . 3  |-  ( v  =  V  ->  ( # `
 ( v  \  U ) )  =  ( # `  ( V  \  U ) ) )
107, 9oveq12d 6295 . 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 6305 . 2  |-  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) )  e.  _V
135, 10, 11, 12ovmpt2 6418 1  |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V
)  =  ( (
# `  ( V  i^i  U ) )  -  ( # `  ( V 
\  U ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757    \ cdif 3410    i^i cin 3412   ifcif 3884   ~Pcpw 3954   class class class wbr 4394    |-> cmpt 4452   `'ccnv 4821   "cima 4825   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7553   supcsup 7933   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    < clt 9657    <_ cle 9658    - cmin 9840    / cdiv 10246   NNcn 10575   ZZcz 10904   ...cfz 11724   #chash 12450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282
This theorem is referenced by:  ballotlemgun  28955  ballotlemfg  28956  ballotlemfrc  28957
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