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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2plord | Structured version Visualization version GIF version | ||
| 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 | 
|---|---|
| rrx2plord.o | ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))} | 
| Ref | Expression | 
|---|---|
| rrx2plord | ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5143 | . . 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 2832 | . . 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 6904 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘1) = (𝑋‘1)) | |
| 6 | fveq1 6904 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦‘1) = (𝑌‘1)) | |
| 7 | 5, 6 | breqan12d 5158 | . . . 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 6904 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥‘2) = (𝑋‘2)) | |
| 10 | fveq1 6904 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦‘2) = (𝑌‘2)) | |
| 11 | 9, 10 | breqan12d 5158 | . . . . 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 5539 | . 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))))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 {copab 5204 ‘cfv 6560 1c1 11157 < clt 11296 2c2 12322 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: rrx2plord1 48647 rrx2plord2 48648 rrx2plordisom 48649 | 
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