Metamath Proof Explorer

Theorem ballotlemi1

Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-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 ballotlemi1 
|- ( ( 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 0re 
 |-  0 e. RR
10 1re 
 |-  1 e. RR
11 9 10 resubcli 
 |-  ( 0 - 1 ) e. RR
12 0lt1 
 |-  0 < 1
13 ltsub23 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ 0 e. RR ) -> ( ( 0 - 1 ) < 0 <-> ( 0 - 0 ) < 1 ) )
14 9 10 9 13 mp3an 
 |-  ( ( 0 - 1 ) < 0 <-> ( 0 - 0 ) < 1 )
15 0m0e0 
 |-  ( 0 - 0 ) = 0
16 15 breq1i 
 |-  ( ( 0 - 0 ) < 1 <-> 0 < 1 )
17 14 16 bitr2i 
 |-  ( 0 < 1 <-> ( 0 - 1 ) < 0 )
18 12 17 mpbi 
 |-  ( 0 - 1 ) < 0
19 11 18 gtneii 
 |-  0 =/= ( 0 - 1 )
20 19 nesymi 
 |-  -. ( 0 - 1 ) = 0
21 eldifi 
 |-  ( C e. ( O \ E ) -> C e. O )
22 1nn 
 |-  1 e. NN
23 22 a1i 
 |-  ( C e. ( O \ E ) -> 1 e. NN )
24 1 2 3 4 5 21 23 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 ) ) ) )
25 24 simpld 
 |-  ( C e. ( O \ E ) -> ( -. 1 e. C -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) ) )
26 25 imp 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( F ` C ) ` 1 ) = ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) )
27 1m1e0 
 |-  ( 1 - 1 ) = 0
28 27 fveq2i 
 |-  ( ( F ` C ) ` ( 1 - 1 ) ) = ( ( F ` C ) ` 0 )
29 28 oveq1i 
 |-  ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( ( ( F ` C ) ` 0 ) - 1 )
30 29 a1i 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( ( F ` C ) ` ( 1 - 1 ) ) - 1 ) = ( ( ( F ` C ) ` 0 ) - 1 ) )
31 1 2 3 4 5 ballotlemfval0 
 |-  ( C e. O -> ( ( F ` C ) ` 0 ) = 0 )
32 21 31 syl 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` 0 ) = 0 )
33 32 adantr 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( F ` C ) ` 0 ) = 0 )
34 33 oveq1d 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( ( F ` C ) ` 0 ) - 1 ) = ( 0 - 1 ) )
35 26 30 34 3eqtrrd 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( 0 - 1 ) = ( ( F ` C ) ` 1 ) )
36 35 eqeq1d 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( ( 0 - 1 ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
37 20 36 mtbii 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> -. ( ( F ` C ) ` 1 ) = 0 )
38 1 2 3 4 5 6 7 8 ballotlemiex 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
39 38 simprd 
 |-  ( C e. ( O \ E ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
40 39 ad2antrr 
 |-  ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` ( I ` C ) ) = 0 )
41 fveqeq2 
 |-  ( ( I ` C ) = 1 -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
42 41 adantl 
 |-  ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( ( F ` C ) ` ( I ` C ) ) = 0 <-> ( ( F ` C ) ` 1 ) = 0 ) )
43 40 42 mpbid 
 |-  ( ( ( C e. ( O \ E ) /\ -. 1 e. C ) /\ ( I ` C ) = 1 ) -> ( ( F ` C ) ` 1 ) = 0 )
44 37 43 mtand 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> -. ( I ` C ) = 1 )
45 44 neqned 
 |-  ( ( C e. ( O \ E ) /\ -. 1 e. C ) -> ( I ` C ) =/= 1 )