Metamath Proof Explorer

Theorem ballotlemfrci

Description: Reverse counting preserves a tie at the first tie. (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 ballotlemfrci 
|- ( C e. ( O \ E ) -> ( ( F ` ( R ` 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 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 1 2 3 4 5 6 7 8 ballotlemiex 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
13 12 simpld 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
14 elfzuz 
 |-  ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) e. ( ZZ>= ` 1 ) )
15 eluzfz2 
 |-  ( ( I ` C ) e. ( ZZ>= ` 1 ) -> ( I ` C ) e. ( 1 ... ( I ` C ) ) )
16 13 14 15 3syl 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( I ` C ) ) )
17 1 2 3 4 5 6 7 8 9 10 11 ballotlemfrc 
 |-  ( ( C e. ( O \ E ) /\ ( I ` C ) e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = ( C .^ ( ( ( S ` C ) ` ( I ` C ) ) ... ( I ` C ) ) ) )
18 16 17 mpdan 
 |-  ( C e. ( O \ E ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = ( C .^ ( ( ( S ` C ) ` ( I ` C ) ) ... ( I ` C ) ) ) )
19 1 2 3 4 5 6 7 8 9 ballotlemsi 
 |-  ( C e. ( O \ E ) -> ( ( S ` C ) ` ( I ` C ) ) = 1 )
20 19 oveq1d 
 |-  ( C e. ( O \ E ) -> ( ( ( S ` C ) ` ( I ` C ) ) ... ( I ` C ) ) = ( 1 ... ( I ` C ) ) )
21 20 oveq2d 
 |-  ( C e. ( O \ E ) -> ( C .^ ( ( ( S ` C ) ` ( I ` C ) ) ... ( I ` C ) ) ) = ( C .^ ( 1 ... ( I ` C ) ) ) )
22 18 21 eqtrd 
 |-  ( C e. ( O \ E ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = ( C .^ ( 1 ... ( I ` C ) ) ) )
23 fz1ssfz0 
 |-  ( 1 ... ( M + N ) ) C_ ( 0 ... ( M + N ) )
24 23 13 sseldi 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ( 0 ... ( M + N ) ) )
25 1 2 3 4 5 6 7 8 9 10 11 ballotlemfg 
 |-  ( ( C e. ( O \ E ) /\ ( I ` C ) e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` ( I ` C ) ) = ( C .^ ( 1 ... ( I ` C ) ) ) )
26 24 25 mpdan 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` ( I ` C ) ) = ( C .^ ( 1 ... ( I ` C ) ) ) )
27 12 simprd 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
28 22 26 27 3eqtr2d 
 |-  ( C e. ( O \ E ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = 0 )