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 18148 | . . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
2 | odutos.d | . . . 4 ⊢ 𝐷 = (ODual‘𝐾) | |
3 | 2 | odupos 18056 | . . 3 ⊢ (𝐾 ∈ Poset → 𝐷 ∈ Poset) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Poset) |
5 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2738 | . . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | 5, 6 | tleile 18149 | . . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
8 | vex 3433 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3433 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | brcnv 5784 | . . . . . . 7 ⊢ (𝑥◡(le‘𝐾)𝑦 ↔ 𝑦(le‘𝐾)𝑥) |
11 | 9, 8 | brcnv 5784 | . . . . . . 7 ⊢ (𝑦◡(le‘𝐾)𝑥 ↔ 𝑥(le‘𝐾)𝑦) |
12 | 10, 11 | orbi12i 912 | . . . . . 6 ⊢ ((𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥) ↔ (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
13 | 7, 12 | sylibr 233 | . . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
14 | 13 | 3com23 1125 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
15 | 14 | 3expb 1119 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
16 | 15 | ralrimivva 3115 | . 2 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
17 | 2, 5 | odubas 18019 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐷) |
18 | 2, 6 | oduleval 18017 | . . 3 ⊢ ◡(le‘𝐾) = (le‘𝐷) |
19 | 17, 18 | istos 18146 | . 2 ⊢ (𝐷 ∈ Toset ↔ (𝐷 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥))) |
20 | 4, 16, 19 | sylanbrc 583 | 1 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5073 ◡ccnv 5583 ‘cfv 6426 Basecbs 16922 lecple 16979 ODualcodu 18014 Posetcpo 18035 Tosetctos 18144 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-dec 12448 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ple 16992 df-odu 18015 df-proset 18023 df-poset 18041 df-toset 18145 |
This theorem is referenced by: ordtrest2NEW 31881 |
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