Theorem cortrcltrcl

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

	Ref	Expression
Assertion	cortrcltrcl	⊢ ( t* ∘ t+ ) = t*

Proof

Step	Hyp	Ref	Expression
1		corcltrcl	⊢ ( r* ∘ t+ ) = t*
2	1	eqcomi	⊢ t* = ( r* ∘ t+ )
3	2	coeq1i	⊢ ( t* ∘ t+ ) = ( ( r* ∘ t+ ) ∘ t+ )
4		coass	⊢ ( ( r* ∘ t+ ) ∘ t+ ) = ( r* ∘ ( t+ ∘ t+ ) )
5		cotrcltrcl	⊢ ( t+ ∘ t+ ) = t+
6	5	coeq2i	⊢ ( r* ∘ ( t+ ∘ t+ ) ) = ( r* ∘ t+ )
7	6 1	eqtri	⊢ ( r* ∘ ( t+ ∘ t+ ) ) = t*
8	4 7	eqtri	⊢ ( ( r* ∘ t+ ) ∘ t+ ) = t*
9	3 8	eqtri	⊢ ( t* ∘ t+ ) = t*