Metamath Proof Explorer

Theorem ballotlemfg

Description: Express the value of ( FC ) in terms of .^ . (Contributed by Thierry Arnoux, 21-Apr-2017)

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 , < ) )
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 ballotlemfg 
|- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( C .^ ( 1 ... J ) ) )

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 ballotth.s 
 |-  S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10 ballotth.r 
 |-  R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11 ballotlemg 
 |-  .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
12 eldifi 
 |-  ( C e. ( O \ E ) -> C e. O )
13 12 adantr 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> C e. O )
14 elfzelz 
 |-  ( J e. ( 0 ... ( M + N ) ) -> J e. ZZ )
15 14 adantl 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> J e. ZZ )
16 1 2 3 4 5 13 15 ballotlemfval 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
17 fzfi 
 |-  ( 1 ... ( M + N ) ) e. Fin
18 1 2 3 ballotlemelo 
 |-  ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
19 18 simplbi 
 |-  ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
20 ssfi 
 |-  ( ( ( 1 ... ( M + N ) ) e. Fin /\ C C_ ( 1 ... ( M + N ) ) ) -> C e. Fin )
21 17 19 20 sylancr 
 |-  ( C e. O -> C e. Fin )
22 13 21 syl 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> C e. Fin )
23 fzfid 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( 1 ... J ) e. Fin )
24 1 2 3 4 5 6 7 8 9 10 11 ballotlemgval 
 |-  ( ( C e. Fin /\ ( 1 ... J ) e. Fin ) -> ( C .^ ( 1 ... J ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
25 22 23 24 syl2anc 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( C .^ ( 1 ... J ) ) = ( ( # ` ( ( 1 ... J ) i^i C ) ) - ( # ` ( ( 1 ... J ) \ C ) ) ) )
26 16 25 eqtr4d 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( C .^ ( 1 ... J ) ) )