Theorem rrx2plord

Description: The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point iff its first coordinate is less than the first coordinate of the other point, or the first coordinates of both points are equal and the second coordinate of the first point is less than the second coordinate of the other point: <. a , b >. <_ <. x , y >. iff ( a < x \/ ( a = x /\ b <_ y ) ) . (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	rrx2plord	⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( 𝑋 𝑂 𝑌 ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) )

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		df-br	⊢ ( 𝑋 𝑂 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝑂 )
3	1	eleq2i	⊢ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝑂 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ) ∧ ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ∨ ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ) ) } )
4	2 3	bitri	⊢ ( 𝑋 𝑂 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ) ∧ ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ∨ ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ) ) } )
5		fveq1	⊢ ( 𝑥 = 𝑋 → ( 𝑥 ‘ 1 ) = ( 𝑋 ‘ 1 ) )
6		fveq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 ‘ 1 ) = ( 𝑌 ‘ 1 ) )
7	5 6	breqan12d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ↔ ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ) )
8	5 6	eqeqan12d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ↔ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) )
9		fveq1	⊢ ( 𝑥 = 𝑋 → ( 𝑥 ‘ 2 ) = ( 𝑋 ‘ 2 ) )
10		fveq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 ‘ 2 ) = ( 𝑌 ‘ 2 ) )
11	9 10	breqan12d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ↔ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) )
12	8 11	anbi12d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) )
13	7 12	orbi12d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ∨ ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ) ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) )
14	13	opelopab2a	⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ) ∧ ( ( 𝑥 ‘ 1 ) < ( 𝑦 ‘ 1 ) ∨ ( ( 𝑥 ‘ 1 ) = ( 𝑦 ‘ 1 ) ∧ ( 𝑥 ‘ 2 ) < ( 𝑦 ‘ 2 ) ) ) ) } ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) )
15	4 14	syl5bb	⊢ ( ( 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ) → ( 𝑋 𝑂 𝑌 ↔ ( ( 𝑋 ‘ 1 ) < ( 𝑌 ‘ 1 ) ∨ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) < ( 𝑌 ‘ 2 ) ) ) ) )