Metamath Proof Explorer

Theorem signstfv

Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-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 ) )
Assertion signstfv 
|- ( F e. Word RR -> ( T ` F ) = ( n e. ( 0 ..^ ( # ` F ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) ) ) )

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 fveq2 
 |-  ( f = F -> ( # ` f ) = ( # ` F ) )
6 5 oveq2d 
 |-  ( f = F -> ( 0 ..^ ( # ` f ) ) = ( 0 ..^ ( # ` F ) ) )
7 simpl 
 |-  ( ( f = F /\ i e. ( 0 ... n ) ) -> f = F )
8 7 fveq1d 
 |-  ( ( f = F /\ i e. ( 0 ... n ) ) -> ( f ` i ) = ( F ` i ) )
9 8 fveq2d 
 |-  ( ( f = F /\ i e. ( 0 ... n ) ) -> ( sgn ` ( f ` i ) ) = ( sgn ` ( F ` i ) ) )
10 9 mpteq2dva 
 |-  ( f = F -> ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) = ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) )
11 10 oveq2d 
 |-  ( f = F -> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) ) = ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) ) )
12 6 11 mpteq12dv 
 |-  ( f = F -> ( n e. ( 0 ..^ ( # ` f ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) ) ) = ( n e. ( 0 ..^ ( # ` F ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) ) ) )
13 ovex 
 |-  ( 0 ..^ ( # ` F ) ) e. _V
14 13 mptex 
 |-  ( n e. ( 0 ..^ ( # ` F ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) ) ) e. _V
15 12 3 14 fvmpt 
 |-  ( F e. Word RR -> ( T ` F ) = ( n e. ( 0 ..^ ( # ` F ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( F ` i ) ) ) ) ) )