Metamath Proof Explorer

Theorem ballotlemiex

Description: Properties of ( IC ) . (Contributed by Thierry Arnoux, 12-Dec-2016) (Revised by AV, 6-Oct-2020)

Ref Expression
Hypotheses 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 ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
Assertion ballotlemiex 
|- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )

Proof

Step Hyp Ref Expression
1 ballotth.m 
 |-  M e. NN
2 ballotth.n 
 |-  N e. NN
3 ballotth.o 
 |-  O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4 ballotth.p 
 |-  P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5 ballotth.f 
 |-  F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6 ballotth.e 
 |-  E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7 ballotth.mgtn 
 |-  N < M
8 ballotth.i 
 |-  I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9 1 2 3 4 5 6 7 8 ballotlemi 
 |-  ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
10 ltso 
 |-  < Or RR
11 10 a1i 
 |-  ( C e. ( O \ E ) -> < Or RR )
12 fzfi 
 |-  ( 1 ... ( M + N ) ) e. Fin
13 ssrab2 
 |-  { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ ( 1 ... ( M + N ) )
14 ssfi 
 |-  ( ( ( 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 )
15 12 13 14 mp2an 
 |-  { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin
16 15 a1i 
 |-  ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } e. Fin )
17 1 2 3 4 5 6 7 ballotlem5 
 |-  ( C e. ( O \ E ) -> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
18 rabn0 
 |-  ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) <-> E. k e. ( 1 ... ( M + N ) ) ( ( F ` C ) ` k ) = 0 )
19 17 18 sylibr 
 |-  ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } =/= (/) )
20 fzssuz 
 |-  ( 1 ... ( M + N ) ) C_ ( ZZ>= ` 1 )
21 uzssz 
 |-  ( ZZ>= ` 1 ) C_ ZZ
22 20 21 sstri 
 |-  ( 1 ... ( M + N ) ) C_ ZZ
23 zssre 
 |-  ZZ C_ RR
24 22 23 sstri 
 |-  ( 1 ... ( M + N ) ) C_ RR
25 13 24 sstri 
 |-  { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR
26 25 a1i 
 |-  ( C e. ( O \ E ) -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } C_ RR )
27 fiinfcl 
 |-  ( ( < 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 ) ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
28 11 16 19 26 27 syl13anc 
 |-  ( C e. ( O \ E ) -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
29 9 28 eqeltrd 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
30 fveqeq2 
 |-  ( k = ( I ` C ) -> ( ( ( F ` C ) ` k ) = 0 <-> ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
31 30 elrab 
 |-  ( ( I ` C ) e. { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } <-> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
32 29 31 sylib 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )