Theorem 3cshw

Description: Cyclically shifting a word three times results in a once cyclically shifted word under certain circumstances. (Contributed by AV, 6-Jun-2018) (Revised by AV, 1-Nov-2018)

	Ref	Expression
Assertion	3cshw	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( W cyclShift N ) = ( ( ( W cyclShift M ) cyclShift N ) cyclShift ( ( # ` W ) - M ) ) )

Proof

Step	Hyp	Ref	Expression
1		2cshwid	|- ( ( W e. Word V /\ M e. ZZ ) -> ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) = W )
2	1	3adant2	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) = W )
3	2	eqcomd	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> W = ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) )
4	3	oveq1d	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( W cyclShift N ) = ( ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) cyclShift N ) )
5		cshwcl	|- ( W e. Word V -> ( W cyclShift M ) e. Word V )
6	5	3ad2ant1	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( W cyclShift M ) e. Word V )
7		lencl	|- ( W e. Word V -> ( # ` W ) e. NN0 )
8	7	nn0zd	|- ( W e. Word V -> ( # ` W ) e. ZZ )
9		zsubcl	|- ( ( ( # ` W ) e. ZZ /\ M e. ZZ ) -> ( ( # ` W ) - M ) e. ZZ )
10	8 9	sylan	|- ( ( W e. Word V /\ M e. ZZ ) -> ( ( # ` W ) - M ) e. ZZ )
11	10	3adant2	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( ( # ` W ) - M ) e. ZZ )
12		simp2	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> N e. ZZ )
13		2cshwcom	|- ( ( ( W cyclShift M ) e. Word V /\ ( ( # ` W ) - M ) e. ZZ /\ N e. ZZ ) -> ( ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) cyclShift N ) = ( ( ( W cyclShift M ) cyclShift N ) cyclShift ( ( # ` W ) - M ) ) )
14	6 11 12 13	syl3anc	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) cyclShift N ) = ( ( ( W cyclShift M ) cyclShift N ) cyclShift ( ( # ` W ) - M ) ) )
15	4 14	eqtrd	|- ( ( W e. Word V /\ N e. ZZ /\ M e. ZZ ) -> ( W cyclShift N ) = ( ( ( W cyclShift M ) cyclShift N ) cyclShift ( ( # ` W ) - M ) ) )