Metamath Proof Explorer


Theorem cortrclrcl

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

Ref Expression
Assertion cortrclrcl ( t* ∘ r* ) = t*

Proof

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