Theorem cortrclrtrcl

Description: The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020)

	Ref	Expression
Assertion	cortrclrtrcl	⊢ ( t* ∘ t* ) = t*

Proof

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