Metamath Proof Explorer

Theorem ballotlem1ri

Description: When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-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 ) )
Assertion ballotlem1ri 
|- ( C e. ( O \ E ) -> ( 1 e. ( R ` C ) <-> ( I ` C ) e. C ) )

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 nnaddcl 
 |-  ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
12 1 2 11 mp2an 
 |-  ( M + N ) e. NN
13 nnuz 
 |-  NN = ( ZZ>= ` 1 )
14 12 13 eleqtri 
 |-  ( M + N ) e. ( ZZ>= ` 1 )
15 eluzfz1 
 |-  ( ( M + N ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( M + N ) ) )
16 14 15 mp1i 
 |-  ( C e. ( O \ E ) -> 1 e. ( 1 ... ( M + N ) ) )
17 1 2 3 4 5 6 7 8 ballotlemiex 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
18 17 simpld 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
19 elfzle1 
 |-  ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> 1 <_ ( I ` C ) )
20 18 19 syl 
 |-  ( C e. ( O \ E ) -> 1 <_ ( I ` C ) )
21 1 2 3 4 5 6 7 8 9 10 ballotlemrv1 
 |-  ( ( C e. ( O \ E ) /\ 1 e. ( 1 ... ( M + N ) ) /\ 1 <_ ( I ` C ) ) -> ( 1 e. ( R ` C ) <-> ( ( ( I ` C ) + 1 ) - 1 ) e. C ) )
22 16 20 21 mpd3an23 
 |-  ( C e. ( O \ E ) -> ( 1 e. ( R ` C ) <-> ( ( ( I ` C ) + 1 ) - 1 ) e. C ) )
23 18 elfzelzd 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
24 23 zcnd 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. CC )
25 1cnd 
 |-  ( C e. ( O \ E ) -> 1 e. CC )
26 24 25 pncand 
 |-  ( C e. ( O \ E ) -> ( ( ( I ` C ) + 1 ) - 1 ) = ( I ` C ) )
27 26 eleq1d 
 |-  ( C e. ( O \ E ) -> ( ( ( ( I ` C ) + 1 ) - 1 ) e. C <-> ( I ` C ) e. C ) )
28 22 27 bitrd 
 |-  ( C e. ( O \ E ) -> ( 1 e. ( R ` C ) <-> ( I ` C ) e. C ) )