Theorem cortrcltrcl

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

	Ref	Expression
Assertion	cortrcltrcl	$$⊢\mathrm{t*}\circ \mathrm{t+}=\mathrm{t*}$$

Proof

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