Step |
Hyp |
Ref |
Expression |
1 |
|
simprl |
|- ( ( N e. NN /\ ( W e. Word V /\ ( # ` W ) = N ) ) -> W e. Word V ) |
2 |
|
eleq1 |
|- ( N = ( # ` W ) -> ( N e. NN <-> ( # ` W ) e. NN ) ) |
3 |
2
|
eqcoms |
|- ( ( # ` W ) = N -> ( N e. NN <-> ( # ` W ) e. NN ) ) |
4 |
3
|
adantl |
|- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( N e. NN <-> ( # ` W ) e. NN ) ) |
5 |
|
wrdfin |
|- ( W e. Word V -> W e. Fin ) |
6 |
|
hashnncl |
|- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) ) |
7 |
5 6
|
syl |
|- ( W e. Word V -> ( ( # ` W ) e. NN <-> W =/= (/) ) ) |
8 |
7
|
biimpd |
|- ( W e. Word V -> ( ( # ` W ) e. NN -> W =/= (/) ) ) |
9 |
8
|
adantr |
|- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( ( # ` W ) e. NN -> W =/= (/) ) ) |
10 |
4 9
|
sylbid |
|- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( N e. NN -> W =/= (/) ) ) |
11 |
10
|
impcom |
|- ( ( N e. NN /\ ( W e. Word V /\ ( # ` W ) = N ) ) -> W =/= (/) ) |
12 |
|
lswcl |
|- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) e. V ) |
13 |
1 11 12
|
syl2anc |
|- ( ( N e. NN /\ ( W e. Word V /\ ( # ` W ) = N ) ) -> ( lastS ` W ) e. V ) |