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Metamath Proof Explorer

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 )