Metamath Proof Explorer

Theorem lenpfxcctswrd

Description: The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018) (Revised by AV, 9-May-2020)

		Ref	Expression
	Assertion	lenpfxcctswrd	\|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) ) = ( # ` W ) )

Proof

Step	Hyp	Ref	Expression
1		pfxcctswrd	\|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) = W )
2	1	fveq2d	\|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) ) = ( # ` W ) )

Step

Hyp

Ref

Expression

pfxcctswrd

 |-  ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) = W )

fveq2d

 |-  ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) ) = ( # ` W ) )