Theorem rrx2plord 50036

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‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))}

Assertion

Ref	Expression
rrx2plord	⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))

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 5107	. . 3 ⊢ (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑂)
2		rrx2plord.o	. . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))}
3	2	eleq2i 2856	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ 𝑂 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))})
4	1, 3	bitri 278	. 2 ⊢ (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))})
5		fveq1 6904	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘0) = (𝑋‘0))
6		fveq1 6904	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦‘0) = (𝑌‘0))
7	5, 6	breqan12d 5122	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘0) < (𝑦‘0) ↔ (𝑋‘0) < (𝑌‘0)))
8	5, 6	eqeqan12d 2777	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘0) = (𝑦‘0) ↔ (𝑋‘0) = (𝑌‘0)))
9		fveq1 6904	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥‘1) = (𝑋‘1))
10		fveq1 6904	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦‘1) = (𝑌‘1))
11	9, 10	breqan12d 5122	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘1) < (𝑦‘1) ↔ (𝑋‘1) < (𝑌‘1)))
12	8, 11	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1)) ↔ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1))))
13	7, 12	orbi12d 932	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))) ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))
14	13	opelopab2a 5513	. 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘0) < (𝑦‘0) ∨ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) < (𝑦‘1))))} ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))
15	4, 14	bitrid 286	1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘0) < (𝑌‘0) ∨ ((𝑋‘0) = (𝑌‘0) ∧ (𝑋‘1) < (𝑌‘1)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   class class class wbr 5106  {copab 5170  ‘cfv 6555  0cc0 11224  1c1 11225   < clt 11367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-iota 6509  df-fv 6563

This theorem is used by:  rrx2plord1  50037  rrx2plord2  50038  rrx2plordisom  50039