Theorem pfxcctswrd

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

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

Proof

Step	Hyp	Ref	Expression
1		lencl	|- ( W e. Word V -> ( # ` W ) e. NN0 )
2		nn0fz0	|- ( ( # ` W ) e. NN0 <-> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
3	1 2	sylib	|- ( W e. Word V -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
4	3	adantr	|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
5		ccatpfx	|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) = ( W prefix ( # ` W ) ) )
6	4 5	mpd3an3	|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) = ( W prefix ( # ` W ) ) )
7		pfxid	|- ( W e. Word V -> ( W prefix ( # ` W ) ) = W )
8	7	adantr	|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( W prefix ( # ` W ) ) = W )
9	6 8	eqtrd	|- ( ( W e. Word V /\ M e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , ( # ` W ) >. ) ) = W )