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Metamath Proof Explorer

Theorem rev0

Description: The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015)

Ref Expression
Assertion rev0 
|- ( reverse ` (/) ) = (/)

Proof

Step Hyp Ref Expression
1 wrd0 
 |-  (/) e. Word _V
2 revlen 
 |-  ( (/) e. Word _V -> ( # ` ( reverse ` (/) ) ) = ( # ` (/) ) )
3 1 2 ax-mp 
 |-  ( # ` ( reverse ` (/) ) ) = ( # ` (/) )
4 hash0 
 |-  ( # ` (/) ) = 0
5 3 4 eqtri 
 |-  ( # ` ( reverse ` (/) ) ) = 0
6 fvex 
 |-  ( reverse ` (/) ) e. _V
7 hasheq0 
 |-  ( ( reverse ` (/) ) e. _V -> ( ( # ` ( reverse ` (/) ) ) = 0 <-> ( reverse ` (/) ) = (/) ) )
8 6 7 ax-mp 
 |-  ( ( # ` ( reverse ` (/) ) ) = 0 <-> ( reverse ` (/) ) = (/) )
9 5 8 mpbi 
 |-  ( reverse ` (/) ) = (/)