Metamath Proof Explorer

Theorem ballotlemsgt1

Description: S maps values less than ( IC ) to values greater than 1. (Contributed by Thierry Arnoux, 28-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 ) ) )
Assertion ballotlemsgt1 
|- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> 1 < ( ( S ` C ) ` 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 elfzelz 
 |-  ( J e. ( 1 ... ( M + N ) ) -> J e. ZZ )
11 10 3ad2ant2 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> J e. ZZ )
12 11 zred 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> J e. RR )
13 1 2 3 4 5 6 7 8 ballotlemiex 
 |-  ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
14 13 simpld 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
15 14 elfzelzd 
 |-  ( C e. ( O \ E ) -> ( I ` C ) e. ZZ )
16 15 3ad2ant1 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( I ` C ) e. ZZ )
17 16 zred 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( I ` C ) e. RR )
18 1red 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> 1 e. RR )
19 17 18 readdcld 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( ( I ` C ) + 1 ) e. RR )
20 simp3 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> J < ( I ` C ) )
21 16 zcnd 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( I ` C ) e. CC )
22 1cnd 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> 1 e. CC )
23 21 22 pncand 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( ( ( I ` C ) + 1 ) - 1 ) = ( I ` C ) )
24 20 23 breqtrrd 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> J < ( ( ( I ` C ) + 1 ) - 1 ) )
25 12 19 18 24 ltsub13d 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> 1 < ( ( ( I ` C ) + 1 ) - J ) )
26 1 2 3 4 5 6 7 8 9 ballotlemsv 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) ) -> ( ( S ` C ) ` J ) = if ( J <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - J ) , J ) )
27 26 3adant3 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( ( S ` C ) ` J ) = if ( J <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - J ) , J ) )
28 12 17 20 ltled 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> J <_ ( I ` C ) )
29 28 iftrued 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> if ( J <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - J ) , J ) = ( ( ( I ` C ) + 1 ) - J ) )
30 27 29 eqtrd 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( ( S ` C ) ` J ) = ( ( ( I ` C ) + 1 ) - J ) )
31 25 30 breqtrrd 
 |-  ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> 1 < ( ( S ` C ) ` J ) )