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Theorem ballotlemsup 28071
Description: The set of zeroes of  F satisfies the conditions to have a supremum (Contributed by Thierry Arnoux, 1-Dec-2016.)
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 ,  `'  <  ) )
Assertion
Ref Expression
ballotlemsup  |-  ( C  e.  ( O  \  E )  ->  E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
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   
k, c, E    i, I    y, c, z, k   
y, C, z    y, F, z    y, M, z   
y, N, z    w, k, y, z, C    w, F    w, M    w, N
Allowed substitution hints:    C( x, c)    P( x, y, z, w, i, k, c)    E( x, y, z, w)    F( x)    I( x, y, z, w, c)    M( x)    N( x)    O( x, y, z, w)

Proof of Theorem ballotlemsup
StepHypRef Expression
1 fzfi 12040 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
2 ssrab2 3580 . . . . . 6  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  ( 1 ... ( M  +  N )
)
3 ssfi 7732 . . . . . 6  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } 
C_  ( 1 ... ( M  +  N
) ) )  ->  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin )
41, 2, 3mp2an 672 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin
54a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin )
6 ballotth.m . . . . . 6  |-  M  e.  NN
7 ballotth.n . . . . . 6  |-  N  e.  NN
8 ballotth.o . . . . . 6  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
9 ballotth.p . . . . . 6  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
10 ballotth.f . . . . . 6  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
11 ballotth.e . . . . . 6  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
12 ballotth.mgtn . . . . . 6  |-  N  < 
M
136, 7, 8, 9, 10, 11, 12ballotlem5 28066 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  E. k  e.  ( 1 ... ( M  +  N )
) ( ( F `
 C ) `  k )  =  0 )
14 rabn0 3800 . . . . 5  |-  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  <->  E. k  e.  ( 1 ... ( M  +  N ) ) ( ( F `  C ) `  k
)  =  0 )
1513, 14sylibr 212 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  =/=  (/) )
16 fzssuz 11715 . . . . . . . 8  |-  ( 1 ... ( M  +  N ) )  C_  ( ZZ>= `  1 )
17 nnuz 11108 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1816, 17sseqtr4i 3532 . . . . . . 7  |-  ( 1 ... ( M  +  N ) )  C_  NN
19 nnssre 10531 . . . . . . 7  |-  NN  C_  RR
2018, 19sstri 3508 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  C_  RR
212, 20sstri 3508 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR
2221a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR )
235, 15, 223jca 1171 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  /\  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR ) )
24 ltso 9656 . . . 4  |-  <  Or  RR
25 cnvso 5539 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
2624, 25mpbi 208 . . 3  |-  `'  <  Or  RR
2723, 26jctil 537 . 2  |-  ( C  e.  ( O  \  E )  ->  ( `'  <  Or  RR  /\  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  /\  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR ) ) )
28 fisup2g 7917 . 2  |-  ( ( `'  <  Or  RR  /\  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  /\  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR ) )  ->  E. z  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
2921sseli 3495 . . . 4  |-  ( z  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  ->  z  e.  RR )
3029anim1i 568 . . 3  |-  ( ( z  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  /\  ( A. w  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  ( w `'  <  z  ->  E. y  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } w `'  <  y ) ) )  -> 
( z  e.  RR  /\  ( A. w  e. 
{ k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) ) )
3130reximi2 2926 . 2  |-  ( E. z  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) )  ->  E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
3227, 28, 313syl 20 1  |-  ( C  e.  ( O  \  E )  ->  E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   {crab 2813    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3780   ~Pcpw 4005   class class class wbr 4442    |-> cmpt 4500    Or wor 4794   `'ccnv 4993   ` cfv 5581  (class class class)co 6277   Fincfn 7508   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    < clt 9619    - cmin 9796    / cdiv 10197   NNcn 10527   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663   #chash 12362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363
This theorem is referenced by:  ballotlemimin  28072  ballotlemfrcn0  28096
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