Metamath Proof Explorer

Theorem ballotlemii

Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-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 , < ) )
Assertion ballotlemii 
|- ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( I ` C ) =/= 1 )

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 1e0p1 
 |-  1 = ( 0 + 1 )
10 ax-1ne0 
 |-  1 =/= 0
11 9 10 eqnetrri 
 |-  ( 0 + 1 ) =/= 0
12 11 neii 
 |-  -. ( 0 + 1 ) = 0
13 eldifi 
 |-  ( C e. ( O \ E ) -> C e. O )
14 1nn 
 |-  1 e. NN
15 14 a1i 
 |-  ( C e. ( O \ E ) -> 1 e. NN )
16 1 2 3 4 5 13 15 ballotlemfp1 
 |-  ( C e. ( O \ E ) -> ( ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) /\ ( 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) ) ) )
17 16 simprd 
 |-  ( C e. ( O \ E ) -> ( 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) ) )
18 17 imp 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) )
19 1m1e0 
 |-  ( 1 - 1 ) = 0
20 19 fveq2i 
 |-  ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
21 20 oveq1i 
 |-  ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) = ( ( ( F ` C ) ` 0 ) + 1 )
22 21 a1i 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( ( ( F ` C ) ` ( 1 - 1 ) ) + 1 ) = ( ( ( F ` C ) ` 0 ) + 1 ) )
23 1 2 3 4 5 ballotlemfval0 
 |-  ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
24 13 23 syl 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` 0 ) = 0 )
25 24 adantr 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( ( F ` C ) ` 0 ) = 0 )
26 25 oveq1d 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( ( ( F ` C ) ` 0 ) + 1 ) = ( 0 + 1 ) )
27 18 22 26 3eqtrrd 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( 0 + 1 ) = ( ( F ` C ) ` 1 ) )
28 27 eqeq1d 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( ( 0 + 1 ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
29 12 28 mtbii 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> -. ( ( F ` C ) ` 1 ) = 0 )
30 1 2 3 4 5 6 7 8 ballotlemiex 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
31 30 simprd 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
32 31 ad2antrr 
 |-  ( ( ( C e. ( O \ E ) /\ 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
33 fveqeq2 
 |-  ( ( I ` C ) = 1 -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
34 33 adantl 
 |-  ( ( ( C e. ( O \ E ) /\ 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
35 32 34 mpbid 
 |-  ( ( ( C e. ( O \ E ) /\ 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` 1 ) = 0 )
36 29 35 mtand 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> -. ( I ` C ) = 1 )
37 36 neqned 
 |-  ( ( C e. ( O \ E ) /\ 1 e. C ) -> ( I ` C ) =/= 1 )