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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > odutos | Structured version Visualization version GIF version |
Description: Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
Ref | Expression |
---|---|
odutos.d | ⊢ 𝐷 = (ODual‘𝐾) |
Ref | Expression |
---|---|
odutos | ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tospos 28989 | . . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
2 | odutos.d | . . . 4 ⊢ 𝐷 = (ODual‘𝐾) | |
3 | 2 | odupos 16958 | . . 3 ⊢ (𝐾 ∈ Poset → 𝐷 ∈ Poset) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Poset) |
5 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2610 | . . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | 5, 6 | tleile 28992 | . . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
8 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | brcnv 5227 | . . . . . . 7 ⊢ (𝑥◡(le‘𝐾)𝑦 ↔ 𝑦(le‘𝐾)𝑥) |
11 | 9, 8 | brcnv 5227 | . . . . . . 7 ⊢ (𝑦◡(le‘𝐾)𝑥 ↔ 𝑥(le‘𝐾)𝑦) |
12 | 10, 11 | orbi12i 542 | . . . . . 6 ⊢ ((𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥) ↔ (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
13 | 7, 12 | sylibr 223 | . . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
14 | 13 | 3com23 1263 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
15 | 14 | 3expb 1258 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
16 | 15 | ralrimivva 2954 | . 2 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
17 | 2, 5 | odubas 16956 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐷) |
18 | 2, 6 | oduleval 16954 | . . 3 ⊢ ◡(le‘𝐾) = (le‘𝐷) |
19 | 17, 18 | istos 16858 | . 2 ⊢ (𝐷 ∈ Toset ↔ (𝐷 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥))) |
20 | 4, 16, 19 | sylanbrc 695 | 1 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ◡ccnv 5037 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 Tosetctos 16856 ODualcodu 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ple 15788 df-preset 16751 df-poset 16769 df-toset 16857 df-odu 16952 |
This theorem is referenced by: ordtrest2NEW 29297 |
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