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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 ) )