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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 30204 | . . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
2 | odutos.d | . . . 4 ⊢ 𝐷 = (ODual‘𝐾) | |
3 | 2 | odupos 17489 | . . 3 ⊢ (𝐾 ∈ Poset → 𝐷 ∈ Poset) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Poset) |
5 | eqid 2826 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2826 | . . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | 5, 6 | tleile 30207 | . . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
8 | vex 3418 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3418 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | brcnv 5538 | . . . . . . 7 ⊢ (𝑥◡(le‘𝐾)𝑦 ↔ 𝑦(le‘𝐾)𝑥) |
11 | 9, 8 | brcnv 5538 | . . . . . . 7 ⊢ (𝑦◡(le‘𝐾)𝑥 ↔ 𝑥(le‘𝐾)𝑦) |
12 | 10, 11 | orbi12i 945 | . . . . . 6 ⊢ ((𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥) ↔ (𝑦(le‘𝐾)𝑥 ∨ 𝑥(le‘𝐾)𝑦)) |
13 | 7, 12 | sylibr 226 | . . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
14 | 13 | 3com23 1162 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
15 | 14 | 3expb 1155 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
16 | 15 | ralrimivva 3181 | . 2 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥)) |
17 | 2, 5 | odubas 17487 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐷) |
18 | 2, 6 | oduleval 17485 | . . 3 ⊢ ◡(le‘𝐾) = (le‘𝐷) |
19 | 17, 18 | istos 17389 | . 2 ⊢ (𝐷 ∈ Toset ↔ (𝐷 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥◡(le‘𝐾)𝑦 ∨ 𝑦◡(le‘𝐾)𝑥))) |
20 | 4, 16, 19 | sylanbrc 580 | 1 ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 880 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3118 class class class wbr 4874 ◡ccnv 5342 ‘cfv 6124 Basecbs 16223 lecple 16313 Posetcpo 17294 Tosetctos 17387 ODualcodu 17482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-dec 11823 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ple 16326 df-proset 17282 df-poset 17300 df-toset 17388 df-odu 17483 |
This theorem is referenced by: ordtrest2NEW 30515 |
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