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Theorem ballotlemsup 28626
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 11985 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
2 ssrab2 3499 . . . . . 6  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  ( 1 ... ( M  +  N )
)
3 ssfi 7656 . . . . . 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 670 . . . . 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 28621 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  E. k  e.  ( 1 ... ( M  +  N )
) ( ( F `
 C ) `  k )  =  0 )
14 rabn0 3732 . . . . 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 11646 . . . . . . . 8  |-  ( 1 ... ( M  +  N ) )  C_  ( ZZ>= `  1 )
17 nnuz 11036 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1816, 17sseqtr4i 3450 . . . . . . 7  |-  ( 1 ... ( M  +  N ) )  C_  NN
19 nnssre 10456 . . . . . . 7  |-  NN  C_  RR
2018, 19sstri 3426 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  C_  RR
212, 20sstri 3426 . . . . 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 1174 . . 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 gtso 9577 . . 3  |-  `'  <  Or  RR
2523, 24jctil 535 . 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 ) ) )
26 fisup2g 7841 . 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
) ) )
2721sseli 3413 . . . 4  |-  ( z  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  ->  z  e.  RR )
2827anim1i 566 . . 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
) ) ) )
2928reximi2 2849 . 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
) ) )
3025, 26, 293syl 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736    \ cdif 3386    i^i cin 3388    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   class class class wbr 4367    |-> cmpt 4425    Or wor 4713   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   Fincfn 7435   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    < clt 9539    - cmin 9718    / cdiv 10123   NNcn 10452   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593   #chash 12307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308
This theorem is referenced by:  ballotlemimin  28627  ballotlemfrcn0  28651
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