Theorem cortrclrcl

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

	Ref	Expression
Assertion	cortrclrcl	$$⊢\mathrm{t*}\circ \mathrm{r*}=\mathrm{t*}$$

Proof

Step	Hyp	Ref	Expression
1		cotrclrcl	$$⊢\mathrm{t+}\circ \mathrm{r*}=\mathrm{t*}$$
2	1	eqcomi	$$⊢\mathrm{t*}=\mathrm{t+}\circ \mathrm{r*}$$
3	2	coeq1i	$$⊢\mathrm{t*}\circ \mathrm{r*}=\left(\mathrm{t+}\circ \mathrm{r*}\right)\circ \mathrm{r*}$$
4		coass	$$⊢\left(\mathrm{t+}\circ \mathrm{r*}\right)\circ \mathrm{r*}=\mathrm{t+}\circ \left(\mathrm{r*}\circ \mathrm{r*}\right)$$
5		corclrcl	$$⊢\mathrm{r*}\circ \mathrm{r*}=\mathrm{r*}$$
6	5	coeq2i	$$⊢\mathrm{t+}\circ \left(\mathrm{r*}\circ \mathrm{r*}\right)=\mathrm{t+}\circ \mathrm{r*}$$
7	6 1	eqtri	$$⊢\mathrm{t+}\circ \left(\mathrm{r*}\circ \mathrm{r*}\right)=\mathrm{t*}$$
8	4 7	eqtri	$$⊢\left(\mathrm{t+}\circ \mathrm{r*}\right)\circ \mathrm{r*}=\mathrm{t*}$$
9	3 8	eqtri	$$⊢\mathrm{t*}\circ \mathrm{r*}=\mathrm{t*}$$