Metamath Proof Explorer

Theorem signshnz

Description: H is not the empty word. (Contributed by Thierry Arnoux, 14-Oct-2018)

Ref Expression
Hypotheses signsv.p 
|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsv.w 
|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
signsv.t 
|- T = ( f e. Word RR |-> ( n e. ( 0 ..^ ( # ` f ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) ) ) )
signsv.v 
|- V = ( f e. Word RR |-> sum_ j e. ( 1 ..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
signs.h 
|- H = ( ( <" 0 "> ++ F ) oF - ( ( F ++ <" 0 "> ) oFC x. C ) )
Assertion signshnz 
|- ( ( F e. Word RR /\ C e. RR+ ) -> H =/= (/) )

Proof

Step Hyp Ref Expression
1 signsv.p 
 |-  .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2 signsv.w 
 |-  W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
3 signsv.t 
 |-  T = ( f e. Word RR |-> ( n e. ( 0 ..^ ( # ` f ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) ) ) )
4 signsv.v 
 |-  V = ( f e. Word RR |-> sum_ j e. ( 1 ..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
5 signs.h 
 |-  H = ( ( <" 0 "> ++ F ) oF - ( ( F ++ <" 0 "> ) oFC x. C ) )
6 1 2 3 4 5 signshlen 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` H ) = ( ( # ` F ) + 1 ) )
7 lencl 
 |-  ( F e. Word RR -> ( # ` F ) e. NN0 )
8 7 adantr 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` F ) e. NN0 )
9 nn0p1nn 
 |-  ( ( # ` F ) e. NN0 -> ( ( # ` F ) + 1 ) e. NN )
10 8 9 syl 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( ( # ` F ) + 1 ) e. NN )
11 6 10 eqeltrd 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` H ) e. NN )
12 11 nnne0d 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` H ) =/= 0 )
13 1 2 3 4 5 signshwrd 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> H e. Word RR )
14 hasheq0 
 |-  ( H e. Word RR -> ( ( # ` H ) = 0 <-> H = (/) ) )
15 13 14 syl 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( ( # ` H ) = 0 <-> H = (/) ) )
16 15 necon3bid 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> ( ( # ` H ) =/= 0 <-> H =/= (/) ) )
17 12 16 mpbid 
 |-  ( ( F e. Word RR /\ C e. RR+ ) -> H =/= (/) )