Description: Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | cortrclrcl | ⊢ ( t* ∘ r* ) = t* |
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* |