Theorem rrx2plord 48667

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.)

Hypothesis

Ref	Expression
rrx2plord.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}

Assertion

Ref	Expression
rrx2plord	⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝑂(𝑥,𝑦)

Proof of Theorem rrx2plord

Step	Hyp	Ref	Expression
1		df-br 5125	. . 3 ⊢ (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑂)
2		rrx2plord.o	. . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}
3	2	eleq2i 2827	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ 𝑂 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))})
4	1, 3	bitri 275	. 2 ⊢ (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))})
5		fveq1 6880	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘1) = (𝑋‘1))
6		fveq1 6880	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦‘1) = (𝑌‘1))
7	5, 6	breqan12d 5140	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘1) < (𝑦‘1) ↔ (𝑋‘1) < (𝑌‘1)))
8	5, 6	eqeqan12d 2750	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘1) = (𝑦‘1) ↔ (𝑋‘1) = (𝑌‘1)))
9		fveq1 6880	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥‘2) = (𝑋‘2))
10		fveq1 6880	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦‘2) = (𝑌‘2))
11	9, 10	breqan12d 5140	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘2) < (𝑦‘2) ↔ (𝑋‘2) < (𝑌‘2)))
12	8, 11	anbi12d 632	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2)) ↔ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))
13	7, 12	orbi12d 918	. . 3 ⊢ ((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 5515	. 2 ⊢ ((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	bitrid 283	1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ⟨cop 4612   class class class wbr 5124  {copab 5186  ‘cfv 6536  1c1 11135   < clt 11274  2c2 12300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-iota 6489  df-fv 6544

This theorem is referenced by:  rrx2plord1  48668  rrx2plord2  48669  rrx2plordisom  48670