Theorem lswccats1

Description: The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018) (Proof shortened by AV, 22-Oct-2018)

	Ref	Expression
Assertion	lswccats1	|- ( ( W e. Word V /\ S e. V ) -> ( lastS ` ( W ++ <" S "> ) ) = S )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( W e. Word V /\ S e. V ) -> W e. Word V )
2		s1cl	|- ( S e. V -> <" S "> e. Word V )
3	2	adantl	|- ( ( W e. Word V /\ S e. V ) -> <" S "> e. Word V )
4		s1nz	|- <" S "> =/= (/)
5	4	a1i	|- ( ( W e. Word V /\ S e. V ) -> <" S "> =/= (/) )
6		lswccatn0lsw	|- ( ( W e. Word V /\ <" S "> e. Word V /\ <" S "> =/= (/) ) -> ( lastS ` ( W ++ <" S "> ) ) = ( lastS ` <" S "> ) )
7	1 3 5 6	syl3anc	|- ( ( W e. Word V /\ S e. V ) -> ( lastS ` ( W ++ <" S "> ) ) = ( lastS ` <" S "> ) )
8		lsws1	|- ( S e. V -> ( lastS ` <" S "> ) = S )
9	8	adantl	|- ( ( W e. Word V /\ S e. V ) -> ( lastS ` <" S "> ) = S )
10	7 9	eqtrd	|- ( ( W e. Word V /\ S e. V ) -> ( lastS ` ( W ++ <" S "> ) ) = S )