Theorem signsvf1

Description: In a single-letter word, which represents a constant polynomial, there is no change of sign. (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	signsvf1	|- ( K e. RR -> ( V ` <" K "> ) = 0 )

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		s1cl	|- ( K e. RR -> <" K "> e. Word RR )
6	1 2 3 4	signsvvfval	|- ( <" K "> e. Word RR -> ( V ` <" K "> ) = sum_ j e. ( 1 ..^ ( # ` <" K "> ) ) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 ) )
7	5 6	syl	|- ( K e. RR -> ( V ` <" K "> ) = sum_ j e. ( 1 ..^ ( # ` <" K "> ) ) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 ) )
8		s1len	|- ( # ` <" K "> ) = 1
9	8	oveq2i	|- ( 1 ..^ ( # ` <" K "> ) ) = ( 1 ..^ 1 )
10		fzo0	|- ( 1 ..^ 1 ) = (/)
11	9 10	eqtri	|- ( 1 ..^ ( # ` <" K "> ) ) = (/)
12	11	sumeq1i	|- sum_ j e. ( 1 ..^ ( # ` <" K "> ) ) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 ) = sum_ j e. (/) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 )
13		sum0	|- sum_ j e. (/) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 ) = 0
14	12 13	eqtri	|- sum_ j e. ( 1 ..^ ( # ` <" K "> ) ) if ( ( ( T ` <" K "> ) ` j ) =/= ( ( T ` <" K "> ) ` ( j - 1 ) ) , 1 , 0 ) = 0
15	7 14	eqtrdi	|- ( K e. RR -> ( V ` <" K "> ) = 0 )