Theorem cotrclrtrcl

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

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

Proof

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