Theorem cshwn

Description: A word cyclically shifted by its length is the word itself. (Contributed by AV, 16-May-2018) (Revised by AV, 20-May-2018) (Revised by AV, 26-Oct-2018)

	Ref	Expression
Assertion	cshwn	|- ( W e. Word V -> ( W cyclShift ( # ` W ) ) = W )

Proof

Step	Hyp	Ref	Expression
1		0csh0	|- ( (/) cyclShift ( # ` W ) ) = (/)
2		oveq1	|- ( (/) = W -> ( (/) cyclShift ( # ` W ) ) = ( W cyclShift ( # ` W ) ) )
3		id	|- ( (/) = W -> (/) = W )
4	1 2 3	3eqtr3a	|- ( (/) = W -> ( W cyclShift ( # ` W ) ) = W )
5	4	a1d	|- ( (/) = W -> ( W e. Word V -> ( W cyclShift ( # ` W ) ) = W ) )
6		lencl	|- ( W e. Word V -> ( # ` W ) e. NN0 )
7	6	nn0zd	|- ( W e. Word V -> ( # ` W ) e. ZZ )
8		cshwmodn	|- ( ( W e. Word V /\ ( # ` W ) e. ZZ ) -> ( W cyclShift ( # ` W ) ) = ( W cyclShift ( ( # ` W ) mod ( # ` W ) ) ) )
9	7 8	mpdan	|- ( W e. Word V -> ( W cyclShift ( # ` W ) ) = ( W cyclShift ( ( # ` W ) mod ( # ` W ) ) ) )
10	9	adantl	|- ( ( (/) =/= W /\ W e. Word V ) -> ( W cyclShift ( # ` W ) ) = ( W cyclShift ( ( # ` W ) mod ( # ` W ) ) ) )
11		necom	|- ( (/) =/= W <-> W =/= (/) )
12		lennncl	|- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
13	11 12	sylan2b	|- ( ( W e. Word V /\ (/) =/= W ) -> ( # ` W ) e. NN )
14	13	nnrpd	|- ( ( W e. Word V /\ (/) =/= W ) -> ( # ` W ) e. RR+ )
15	14	ancoms	|- ( ( (/) =/= W /\ W e. Word V ) -> ( # ` W ) e. RR+ )
16		modid0	|- ( ( # ` W ) e. RR+ -> ( ( # ` W ) mod ( # ` W ) ) = 0 )
17	15 16	syl	|- ( ( (/) =/= W /\ W e. Word V ) -> ( ( # ` W ) mod ( # ` W ) ) = 0 )
18	17	oveq2d	|- ( ( (/) =/= W /\ W e. Word V ) -> ( W cyclShift ( ( # ` W ) mod ( # ` W ) ) ) = ( W cyclShift 0 ) )
19		cshw0	|- ( W e. Word V -> ( W cyclShift 0 ) = W )
20	19	adantl	|- ( ( (/) =/= W /\ W e. Word V ) -> ( W cyclShift 0 ) = W )
21	10 18 20	3eqtrd	|- ( ( (/) =/= W /\ W e. Word V ) -> ( W cyclShift ( # ` W ) ) = W )
22	21	ex	|- ( (/) =/= W -> ( W e. Word V -> ( W cyclShift ( # ` W ) ) = W ) )
23	5 22	pm2.61ine	|- ( W e. Word V -> ( W cyclShift ( # ` W ) ) = W )