Metamath Proof Explorer


Theorem rrx2plord1

Description: The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point if its first coordinate is less than the first coordinate of the other point. (Contributed by AV, 12-Mar-2023)

Ref Expression
Hypothesis rrx2plord.o O = x y | x R y R x 1 < y 1 x 1 = y 1 x 2 < y 2
Assertion rrx2plord1 X R Y R X 1 < Y 1 X O Y

Proof

Step Hyp Ref Expression
1 rrx2plord.o O = x y | x R y R x 1 < y 1 x 1 = y 1 x 2 < y 2
2 simp3 X R Y R X 1 < Y 1 X 1 < Y 1
3 2 orcd X R Y R X 1 < Y 1 X 1 < Y 1 X 1 = Y 1 X 2 < Y 2
4 1 rrx2plord X R Y R X O Y X 1 < Y 1 X 1 = Y 1 X 2 < Y 2
5 4 3adant3 X R Y R X 1 < Y 1 X O Y X 1 < Y 1 X 1 = Y 1 X 2 < Y 2
6 3 5 mpbird X R Y R X 1 < Y 1 X O Y