Metamath Proof Explorer

Theorem cshwidx0

Description: The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N of the original word. (Contributed by AV, 15-May-2018) (Revised by AV, 21-May-2018) (Revised by AV, 30-Oct-2018)

Ref Expression
Assertion cshwidx0 
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) )

Proof

Step Hyp Ref Expression
1 hasheq0 
 |-  ( W e. Word V -> ( ( # ` W ) = 0 <-> W = (/) ) )
2 elfzo0 
 |-  ( N e. ( 0 ..^ ( # ` W ) ) <-> ( N e. NN0 /\ ( # ` W ) e. NN /\ N < ( # ` W ) ) )
3 elnnne0 
 |-  ( ( # ` W ) e. NN <-> ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) )
4 eqneqall 
 |-  ( ( # ` W ) = 0 -> ( ( # ` W ) =/= 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
5 4 com12 
 |-  ( ( # ` W ) =/= 0 -> ( ( # ` W ) = 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
6 5 adantl 
 |-  ( ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) -> ( ( # ` W ) = 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
7 3 6 sylbi 
 |-  ( ( # ` W ) e. NN -> ( ( # ` W ) = 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
8 7 3ad2ant2 
 |-  ( ( N e. NN0 /\ ( # ` W ) e. NN /\ N < ( # ` W ) ) -> ( ( # ` W ) = 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
9 2 8 sylbi 
 |-  ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( # ` W ) = 0 -> ( W e. Word V -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
10 9 com13 
 |-  ( W e. Word V -> ( ( # ` W ) = 0 -> ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
11 1 10 sylbird 
 |-  ( W e. Word V -> ( W = (/) -> ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
12 11 com23 
 |-  ( W e. Word V -> ( N e. ( 0 ..^ ( # ` W ) ) -> ( W = (/) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) ) )
13 12 imp 
 |-  ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( W = (/) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) )
14 13 com12 
 |-  ( W = (/) -> ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) )
15 simpl 
 |-  ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> W e. Word V )
16 15 adantl 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> W e. Word V )
17 simpl 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> W =/= (/) )
18 elfzoelz 
 |-  ( N e. ( 0 ..^ ( # ` W ) ) -> N e. ZZ )
19 18 ad2antll 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> N e. ZZ )
20 cshwidx0mod 
 |-  ( ( W e. Word V /\ W =/= (/) /\ N e. ZZ ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` ( N mod ( # ` W ) ) ) )
21 16 17 19 20 syl3anc 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` ( N mod ( # ` W ) ) ) )
22 zmodidfzoimp 
 |-  ( N e. ( 0 ..^ ( # ` W ) ) -> ( N mod ( # ` W ) ) = N )
23 22 ad2antll 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> ( N mod ( # ` W ) ) = N )
24 23 fveq2d 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> ( W ` ( N mod ( # ` W ) ) ) = ( W ` N ) )
25 21 24 eqtrd 
 |-  ( ( W =/= (/) /\ ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) )
26 25 ex 
 |-  ( W =/= (/) -> ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) ) )
27 14 26 pm2.61ine 
 |-  ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) )