Theorem lswcl

Description: Closure of the last symbol: the last symbol of a not empty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018) (Proof shortened by AV, 29-Apr-2020)

	Ref	Expression
Assertion	lswcl	|- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) e. V )

Proof

Step	Hyp	Ref	Expression
1		lsw	|- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
2	1	adantr	|- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
3		lennncl	|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
4		fzo0end	|- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) )
5	3 4	syl	|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) )
6		wrdsymbcl	|- ( ( W e. Word V /\ ( ( # ` W ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( # ` W ) - 1 ) ) e. V )
7	5 6	syldan	|- ( ( W e. Word V /\ W =/= (/) ) -> ( W ` ( ( # ` W ) - 1 ) ) e. V )
8	2 7	eqeltrd	|- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) e. V )