Theorem ccatfv0

Description: The first symbol of a concatenation of two words is the first symbol of the first word if the first word is not empty. (Contributed by Alexander van der Vekens, 22-Sep-2018)

	Ref	Expression
Assertion	ccatfv0	|- ( ( A e. Word V /\ B e. Word V /\ 0 < ( # ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )

Proof

Step	Hyp	Ref	Expression
1		lencl	|- ( A e. Word V -> ( # ` A ) e. NN0 )
2		elnnnn0b	|- ( ( # ` A ) e. NN <-> ( ( # ` A ) e. NN0 /\ 0 < ( # ` A ) ) )
3	2	biimpri	|- ( ( ( # ` A ) e. NN0 /\ 0 < ( # ` A ) ) -> ( # ` A ) e. NN )
4	1 3	sylan	|- ( ( A e. Word V /\ 0 < ( # ` A ) ) -> ( # ` A ) e. NN )
5		lbfzo0	|- ( 0 e. ( 0 ..^ ( # ` A ) ) <-> ( # ` A ) e. NN )
6	4 5	sylibr	|- ( ( A e. Word V /\ 0 < ( # ` A ) ) -> 0 e. ( 0 ..^ ( # ` A ) ) )
7	6	3adant2	|- ( ( A e. Word V /\ B e. Word V /\ 0 < ( # ` A ) ) -> 0 e. ( 0 ..^ ( # ` A ) ) )
8		ccatval1	|- ( ( A e. Word V /\ B e. Word V /\ 0 e. ( 0 ..^ ( # ` A ) ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
9	7 8	syld3an3	|- ( ( A e. Word V /\ B e. Word V /\ 0 < ( # ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )