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Theorem ballotlemieq 28961
Description: If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-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 ) ) )
Assertion
Ref Expression
ballotlemieq  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( S `
 D ) )
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    C, i, k    i, E, k    C, k    k, I, c    E, c    i, I, c    S, k    D, i, k
Allowed substitution hints:    C( x, c)    D( x, c)    P( x, i, k, c)    S( x, i, c)    E( x)    F( x)    I( x)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemieq
StepHypRef Expression
1 simpl 455 . . . . . 6  |-  ( ( ( I `  C
)  =  ( I `
 D )  /\  i  e.  ( 1 ... ( M  +  N ) ) )  ->  ( I `  C )  =  ( I `  D ) )
21breq2d 4407 . . . . 5  |-  ( ( ( I `  C
)  =  ( I `
 D )  /\  i  e.  ( 1 ... ( M  +  N ) ) )  ->  ( i  <_ 
( I `  C
)  <->  i  <_  (
I `  D )
) )
31oveq1d 6293 . . . . . 6  |-  ( ( ( I `  C
)  =  ( I `
 D )  /\  i  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( I `
 C )  +  1 )  =  ( ( I `  D
)  +  1 ) )
43oveq1d 6293 . . . . 5  |-  ( ( ( I `  C
)  =  ( I `
 D )  /\  i  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( ( I `  C )  +  1 )  -  i )  =  ( ( ( I `  D )  +  1 )  -  i ) )
52, 4ifbieq1d 3908 . . . 4  |-  ( ( ( I `  C
)  =  ( I `
 D )  /\  i  e.  ( 1 ... ( M  +  N ) ) )  ->  if ( i  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  i ) ,  i )  =  if ( i  <_ 
( I `  D
) ,  ( ( ( I `  D
)  +  1 )  -  i ) ,  i ) )
65mpteq2dva 4481 . . 3  |-  ( ( I `  C )  =  ( I `  D )  ->  (
i  e.  ( 1 ... ( M  +  N ) )  |->  if ( i  <_  (
I `  C ) ,  ( ( ( I `  C )  +  1 )  -  i ) ,  i ) )  =  ( i  e.  ( 1 ... ( M  +  N ) )  |->  if ( i  <_  (
I `  D ) ,  ( ( ( I `  D )  +  1 )  -  i ) ,  i ) ) )
763ad2ant3 1020 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  C
) ,  ( ( ( I `  C
)  +  1 )  -  i ) ,  i ) )  =  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  D
) ,  ( ( ( I `  D
)  +  1 )  -  i ) ,  i ) ) )
8 ballotth.m . . . 4  |-  M  e.  NN
9 ballotth.n . . . 4  |-  N  e.  NN
10 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
11 ballotth.p . . . 4  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
12 ballotth.f . . . 4  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
13 ballotth.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
14 ballotth.mgtn . . . 4  |-  N  < 
M
15 ballotth.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
16 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
178, 9, 10, 11, 12, 13, 14, 15, 16ballotlemsval 28953 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( S `  C )  =  ( i  e.  ( 1 ... ( M  +  N )
)  |->  if ( i  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  i ) ,  i ) ) )
18173ad2ant1 1018 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( i  e.  ( 1 ... ( M  +  N
) )  |->  if ( i  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  i ) ,  i ) ) )
198, 9, 10, 11, 12, 13, 14, 15, 16ballotlemsval 28953 . . 3  |-  ( D  e.  ( O  \  E )  ->  ( S `  D )  =  ( i  e.  ( 1 ... ( M  +  N )
)  |->  if ( i  <_  ( I `  D ) ,  ( ( ( I `  D )  +  1 )  -  i ) ,  i ) ) )
20193ad2ant2 1019 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  D
)  =  ( i  e.  ( 1 ... ( M  +  N
) )  |->  if ( i  <_  ( I `  D ) ,  ( ( ( I `  D )  +  1 )  -  i ) ,  i ) ) )
217, 18, 203eqtr4d 2453 1  |-  ( ( C  e.  ( O 
\  E )  /\  D  e.  ( O  \  E )  /\  (
I `  C )  =  ( I `  D ) )  -> 
( S `  C
)  =  ( S `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   {crab 2758    \ cdif 3411    i^i cin 3413   ifcif 3885   ~Pcpw 3955   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4822   ` cfv 5569  (class class class)co 6278   supcsup 7934   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    < clt 9658    <_ cle 9659    - cmin 9841    / cdiv 10247   NNcn 10576   ZZcz 10905   ...cfz 11726   #chash 12452
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-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
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 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281
This theorem is referenced by:  ballotlemrinv0  28977
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