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 e. R /\ y e. R ) /\ ( ( x ` 1 ) < ( y ` 1 ) \/ ( ( x ` 1 ) = ( y ` 1 ) /\ ( x ` 2 ) < ( y ` 2 ) ) ) ) }
Assertion	rrx2plord1	|- ( ( X e. R /\ Y e. R /\ ( X ` 1 ) < ( Y ` 1 ) ) -> X O Y )

Proof

Step	Hyp	Ref	Expression
1		rrx2plord.o	|- O = { <. x , y >. | ( ( x e. R /\ y e. R ) /\ ( ( x ` 1 ) < ( y ` 1 ) \/ ( ( x ` 1 ) = ( y ` 1 ) /\ ( x ` 2 ) < ( y ` 2 ) ) ) ) }
2		simp3	|- ( ( X e. R /\ Y e. R /\ ( X ` 1 ) < ( Y ` 1 ) ) -> ( X ` 1 ) < ( Y ` 1 ) )
3	2	orcd	|- ( ( X e. R /\ Y e. R /\ ( X ` 1 ) < ( Y ` 1 ) ) -> ( ( X ` 1 ) < ( Y ` 1 ) \/ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) < ( Y ` 2 ) ) ) )
4	1	rrx2plord	|- ( ( X e. R /\ Y e. R ) -> ( X O Y <-> ( ( X ` 1 ) < ( Y ` 1 ) \/ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) < ( Y ` 2 ) ) ) ) )
5	4	3adant3	|- ( ( X e. R /\ Y e. 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 e. R /\ Y e. R /\ ( X ` 1 ) < ( Y ` 1 ) ) -> X O Y )