Metamath Proof Explorer


Theorem ttrclco

Description: Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024)

Ref Expression
Assertion ttrclco t++ R R t++ R

Proof

Step Hyp Ref Expression
1 relres Rel R V
2 ssttrcl Rel R V R V t++ R V
3 coss2 R V t++ R V t++ R V R V t++ R V t++ R V
4 1 2 3 mp2b t++ R V R V t++ R V t++ R V
5 ttrcltr t++ R V t++ R V t++ R V
6 4 5 sstri t++ R V R V t++ R V
7 relco Rel t++ R V R
8 dfrel3 Rel t++ R V R t++ R V R V = t++ R V R
9 7 8 mpbi t++ R V R V = t++ R V R
10 resco t++ R V R V = t++ R V R V
11 ttrclresv t++ R V = t++ R
12 11 coeq1i t++ R V R = t++ R R
13 9 10 12 3eqtr3i t++ R V R V = t++ R R
14 6 13 11 3sstr3i t++ R R t++ R