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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2plord2 | Structured version Visualization version GIF version |
Description: The lexicographical ordering for points in the two dimensional Euclidean plane: if the first coordinates of two points are equal, a point is less than another point iff the second coordinate of the point is less than the second coordinate of the other point. (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))))} |
rrx2plord2.r | ⊢ 𝑅 = (ℝ ↑m {1, 2}) |
Ref | Expression |
---|---|
rrx2plord2 | ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2plord.o | . . . 4 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))} | |
2 | 1 | rrx2plord 47406 | . . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))) |
3 | 2 | 3adant3 1133 | . 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))) |
4 | eqid 2733 | . . . . . . . . . . . 12 ⊢ {1, 2} = {1, 2} | |
5 | rrx2plord2.r | . . . . . . . . . . . 12 ⊢ 𝑅 = (ℝ ↑m {1, 2}) | |
6 | 4, 5 | rrx2pxel 47397 | . . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑅 → (𝑋‘1) ∈ ℝ) |
7 | 6 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋‘1) ∈ ℝ) |
8 | ltne 11311 | . . . . . . . . . . 11 ⊢ (((𝑋‘1) ∈ ℝ ∧ (𝑋‘1) < (𝑌‘1)) → (𝑌‘1) ≠ (𝑋‘1)) | |
9 | 8 | necomd 2997 | . . . . . . . . . 10 ⊢ (((𝑋‘1) ∈ ℝ ∧ (𝑋‘1) < (𝑌‘1)) → (𝑋‘1) ≠ (𝑌‘1)) |
10 | 7, 9 | sylan 581 | . . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) ∧ (𝑋‘1) < (𝑌‘1)) → (𝑋‘1) ≠ (𝑌‘1)) |
11 | 10 | ex 414 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → ((𝑋‘1) < (𝑌‘1) → (𝑋‘1) ≠ (𝑌‘1))) |
12 | eqneqall 2952 | . . . . . . . 8 ⊢ ((𝑋‘1) = (𝑌‘1) → ((𝑋‘1) ≠ (𝑌‘1) → (𝑋‘2) < (𝑌‘2))) | |
13 | 11, 12 | syl9 77 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → ((𝑋‘1) = (𝑌‘1) → ((𝑋‘1) < (𝑌‘1) → (𝑋‘2) < (𝑌‘2)))) |
14 | 13 | 3impia 1118 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → ((𝑋‘1) < (𝑌‘1) → (𝑋‘2) < (𝑌‘2))) |
15 | 14 | com12 32 | . . . . 5 ⊢ ((𝑋‘1) < (𝑌‘1) → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋‘2) < (𝑌‘2))) |
16 | simpr 486 | . . . . . 6 ⊢ (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)) → (𝑋‘2) < (𝑌‘2)) | |
17 | 16 | a1d 25 | . . . . 5 ⊢ (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)) → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋‘2) < (𝑌‘2))) |
18 | 15, 17 | jaoi 856 | . . . 4 ⊢ (((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))) → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋‘2) < (𝑌‘2))) |
19 | 18 | com12 32 | . . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))) → (𝑋‘2) < (𝑌‘2))) |
20 | olc 867 | . . . . 5 ⊢ (((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)) → ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))) | |
21 | 20 | ex 414 | . . . 4 ⊢ ((𝑋‘1) = (𝑌‘1) → ((𝑋‘2) < (𝑌‘2) → ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))) |
22 | 21 | 3ad2ant3 1136 | . . 3 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → ((𝑋‘2) < (𝑌‘2) → ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))))) |
23 | 19, 22 | impbid 211 | . 2 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2))) ↔ (𝑋‘2) < (𝑌‘2))) |
24 | 3, 23 | bitrd 279 | 1 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {cpr 4631 class class class wbr 5149 {copab 5211 ‘cfv 6544 (class class class)co 7409 ↑m cmap 8820 ℝcr 11109 1c1 11111 < clt 11248 2c2 12267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 |
This theorem is referenced by: (None) |
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