Metamath Proof Explorer


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 ) )