Theorem signstfvcl

Description: Closure of the zero skipping sign in case the first letter is not zero. (Contributed by Thierry Arnoux, 10-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	signstfvcl	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) e. { -u 1 , 1 } )

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		simpll	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> F e. ( Word RR \ { (/) } ) )
6	5	eldifad	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> F e. Word RR )
7	1 2 3 4	signstcl	|- ( ( F e. Word RR /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) e. { -u 1 , 0 , 1 } )
8	6 7	sylancom	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) e. { -u 1 , 0 , 1 } )
9	1 2 3 4	signstfvneq0	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) =/= 0 )
10		eldifsn	|- ( ( ( T ` F ) ` N ) e. ( { -u 1 , 0 , 1 } \ { 0 } ) <-> ( ( ( T ` F ) ` N ) e. { -u 1 , 0 , 1 } /\ ( ( T ` F ) ` N ) =/= 0 ) )
11	8 9 10	sylanbrc	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) e. ( { -u 1 , 0 , 1 } \ { 0 } ) )
12		tpcomb	|- { -u 1 , 0 , 1 } = { -u 1 , 1 , 0 }
13	12	difeq1i	|- ( { -u 1 , 0 , 1 } \ { 0 } ) = ( { -u 1 , 1 , 0 } \ { 0 } )
14		neg1ne0	|- -u 1 =/= 0
15		ax-1ne0	|- 1 =/= 0
16		diftpsn3	|- ( ( -u 1 =/= 0 /\ 1 =/= 0 ) -> ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 } )
17	14 15 16	mp2an	|- ( { -u 1 , 1 , 0 } \ { 0 } ) = { -u 1 , 1 }
18	13 17	eqtri	|- ( { -u 1 , 0 , 1 } \ { 0 } ) = { -u 1 , 1 }
19	11 18	eleqtrdi	|- ( ( ( F e. ( Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) e. { -u 1 , 1 } )