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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2plordso | Structured version Visualization version GIF version |
Description: The lexicographical ordering for points in the two dimensional Euclidean plane is a strict total ordering. (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 |
---|---|
rrx2plordso | ⊢ 𝑂 Or 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10709 | . . 3 ⊢ < Or ℝ | |
2 | eqid 2818 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} | |
3 | 2 | soxp 7812 | . . 3 ⊢ (( < Or ℝ ∧ < Or ℝ) → {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} Or (ℝ × ℝ)) |
4 | 1, 1, 3 | mp2an 688 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} Or (ℝ × ℝ) |
5 | rrx2plord.o | . . . 4 ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))} | |
6 | rrx2plord2.r | . . . 4 ⊢ 𝑅 = (ℝ ↑m {1, 2}) | |
7 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {〈1, 𝑥〉, 〈2, 𝑦〉}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {〈1, 𝑥〉, 〈2, 𝑦〉}) | |
8 | 5, 6, 7, 2 | rrx2plordisom 44638 | . . 3 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {〈1, 𝑥〉, 〈2, 𝑦〉}) Isom {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) |
9 | isoso 7090 | . . 3 ⊢ ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {〈1, 𝑥〉, 〈2, 𝑦〉}) Isom {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}, 𝑂((ℝ × ℝ), 𝑅) → ({〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅)) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))} Or (ℝ × ℝ) ↔ 𝑂 Or 𝑅) |
11 | 4, 10 | mpbi 231 | 1 ⊢ 𝑂 Or 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 {cpr 4559 〈cop 4563 class class class wbr 5057 {copab 5119 Or wor 5466 × cxp 5546 ‘cfv 6348 Isom wiso 6349 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7676 2nd c2nd 7677 ↑m cmap 8395 ℝcr 10524 1c1 10526 < clt 10663 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-2 11688 |
This theorem is referenced by: (None) |
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