Step |
Hyp |
Ref |
Expression |
1 |
|
pfxval |
|- ( ( S e. Word A /\ L e. NN0 ) -> ( S prefix L ) = ( S substr <. 0 , L >. ) ) |
2 |
|
simpr |
|- ( ( S e. _V /\ L e. NN0 ) -> L e. NN0 ) |
3 |
2
|
con3i |
|- ( -. L e. NN0 -> -. ( S e. _V /\ L e. NN0 ) ) |
4 |
3
|
adantl |
|- ( ( S e. Word A /\ -. L e. NN0 ) -> -. ( S e. _V /\ L e. NN0 ) ) |
5 |
|
pfxnndmnd |
|- ( -. ( S e. _V /\ L e. NN0 ) -> ( S prefix L ) = (/) ) |
6 |
4 5
|
syl |
|- ( ( S e. Word A /\ -. L e. NN0 ) -> ( S prefix L ) = (/) ) |
7 |
|
simpr |
|- ( ( 0 e. NN0 /\ L e. NN0 ) -> L e. NN0 ) |
8 |
7
|
con3i |
|- ( -. L e. NN0 -> -. ( 0 e. NN0 /\ L e. NN0 ) ) |
9 |
|
swrdnnn0nd |
|- ( ( S e. Word A /\ -. ( 0 e. NN0 /\ L e. NN0 ) ) -> ( S substr <. 0 , L >. ) = (/) ) |
10 |
8 9
|
sylan2 |
|- ( ( S e. Word A /\ -. L e. NN0 ) -> ( S substr <. 0 , L >. ) = (/) ) |
11 |
6 10
|
eqtr4d |
|- ( ( S e. Word A /\ -. L e. NN0 ) -> ( S prefix L ) = ( S substr <. 0 , L >. ) ) |
12 |
1 11
|
pm2.61dan |
|- ( S e. Word A -> ( S prefix L ) = ( S substr <. 0 , L >. ) ) |