Theorem rrx2plord 49754

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 5103	. . 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 2852	. . 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 278	. 2 ⊢ (𝑋𝑂𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))})
5		fveq1 6872	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘1) = (𝑋‘1))
6		fveq1 6872	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦‘1) = (𝑌‘1))
7	5, 6	breqan12d 5118	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘1) < (𝑦‘1) ↔ (𝑋‘1) < (𝑌‘1)))
8	5, 6	eqeqan12d 2774	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘1) = (𝑦‘1) ↔ (𝑋‘1) = (𝑌‘1)))
9		fveq1 6872	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥‘2) = (𝑋‘2))
10		fveq1 6872	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦‘2) = (𝑌‘2))
11	9, 10	breqan12d 5118	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥‘2) < (𝑦‘2) ↔ (𝑋‘2) < (𝑌‘2)))
12	8, 11	anbi12d 644	. . . 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 932	. . 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 5505	. 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 286	1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ⟨cop 4589   class class class wbr 5102  {copab 5166  ‘cfv 6527  1c1 11172   < clt 11314  2c2 12366

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6483  df-fv 6535

This theorem is used by:  rrx2plord1  49755  rrx2plord2  49756  rrx2plordisom  49757