Metamath Proof Explorer

Theorem repswlsw

Description: The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018)

Ref Expression
Assertion repswlsw 
|- ( ( S e. V /\ N e. NN ) -> ( lastS ` ( S repeatS N ) ) = S )

Proof

Step Hyp Ref Expression
1 nnnn0 
 |-  ( N e. NN -> N e. NN0 )
2 repsw 
 |-  ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) e. Word V )
3 1 2 sylan2 
 |-  ( ( S e. V /\ N e. NN ) -> ( S repeatS N ) e. Word V )
4 lsw 
 |-  ( ( S repeatS N ) e. Word V -> ( lastS ` ( S repeatS N ) ) = ( ( S repeatS N ) ` ( ( # ` ( S repeatS N ) ) - 1 ) ) )
5 3 4 syl 
 |-  ( ( S e. V /\ N e. NN ) -> ( lastS ` ( S repeatS N ) ) = ( ( S repeatS N ) ` ( ( # ` ( S repeatS N ) ) - 1 ) ) )
6 simpl 
 |-  ( ( S e. V /\ N e. NN ) -> S e. V )
7 1 adantl 
 |-  ( ( S e. V /\ N e. NN ) -> N e. NN0 )
8 repswlen 
 |-  ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
9 1 8 sylan2 
 |-  ( ( S e. V /\ N e. NN ) -> ( # ` ( S repeatS N ) ) = N )
10 9 oveq1d 
 |-  ( ( S e. V /\ N e. NN ) -> ( ( # ` ( S repeatS N ) ) - 1 ) = ( N - 1 ) )
11 fzo0end 
 |-  ( N e. NN -> ( N - 1 ) e. ( 0 ..^ N ) )
12 11 adantl 
 |-  ( ( S e. V /\ N e. NN ) -> ( N - 1 ) e. ( 0 ..^ N ) )
13 10 12 eqeltrd 
 |-  ( ( S e. V /\ N e. NN ) -> ( ( # ` ( S repeatS N ) ) - 1 ) e. ( 0 ..^ N ) )
14 repswsymb 
 |-  ( ( S e. V /\ N e. NN0 /\ ( ( # ` ( S repeatS N ) ) - 1 ) e. ( 0 ..^ N ) ) -> ( ( S repeatS N ) ` ( ( # ` ( S repeatS N ) ) - 1 ) ) = S )
15 6 7 13 14 syl3anc 
 |-  ( ( S e. V /\ N e. NN ) -> ( ( S repeatS N ) ` ( ( # ` ( S repeatS N ) ) - 1 ) ) = S )
16 5 15 eqtrd 
 |-  ( ( S e. V /\ N e. NN ) -> ( lastS ` ( S repeatS N ) ) = S )