Theorem cotrclrtrcl

Description: Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020)

	Ref	Expression
Assertion	cotrclrtrcl	|- ( t+ o. t* ) = t*

Proof

Step	Hyp	Ref	Expression
1		cotrclrcl	|- ( t+ o. r* ) = t*
2	1	eqcomi	|- t* = ( t+ o. r* )
3	2	coeq2i	|- ( t+ o. t* ) = ( t+ o. ( t+ o. r* ) )
4		coass	|- ( ( t+ o. t+ ) o. r* ) = ( t+ o. ( t+ o. r* ) )
5	4	eqcomi	|- ( t+ o. ( t+ o. r* ) ) = ( ( t+ o. t+ ) o. r* )
6		cotrcltrcl	|- ( t+ o. t+ ) = t+
7	6	coeq1i	|- ( ( t+ o. t+ ) o. r* ) = ( t+ o. r* )
8	7 1	eqtri	|- ( ( t+ o. t+ ) o. r* ) = t*
9	5 8	eqtri	|- ( t+ o. ( t+ o. r* ) ) = t*
10	3 9	eqtri	|- ( t+ o. t* ) = t*