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Mirrors > Home > MPE Home > Th. List > oduposb | Structured version Visualization version GIF version |
Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
odupos.d | ⊢ 𝐷 = (ODual‘𝑂) |
Ref | Expression |
---|---|
oduposb | ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odupos.d | . . 3 ⊢ 𝐷 = (ODual‘𝑂) | |
2 | 1 | odupos 18386 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
3 | eqid 2735 | . . . 4 ⊢ (ODual‘𝐷) = (ODual‘𝐷) | |
4 | 3 | odupos 18386 | . . 3 ⊢ (𝐷 ∈ Poset → (ODual‘𝐷) ∈ Poset) |
5 | fvexd 6922 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (ODual‘𝐷) ∈ V) | |
6 | id 22 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
7 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
8 | 1, 7 | odubas 18348 | . . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐷) |
9 | 3, 8 | odubas 18348 | . . . . 5 ⊢ (Base‘𝑂) = (Base‘(ODual‘𝐷)) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘(ODual‘𝐷))) |
11 | eqidd 2736 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘𝑂)) | |
12 | eqid 2735 | . . . . . . . . . 10 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 12 | oduleval 18346 | . . . . . . . . 9 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
14 | 3, 13 | oduleval 18346 | . . . . . . . 8 ⊢ ◡◡(le‘𝑂) = (le‘(ODual‘𝐷)) |
15 | 14 | eqcomi 2744 | . . . . . . 7 ⊢ (le‘(ODual‘𝐷)) = ◡◡(le‘𝑂) |
16 | 15 | breqi 5154 | . . . . . 6 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎◡◡(le‘𝑂)𝑏) |
17 | vex 3482 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | vex 3482 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | brcnv 5896 | . . . . . 6 ⊢ (𝑎◡◡(le‘𝑂)𝑏 ↔ 𝑏◡(le‘𝑂)𝑎) |
20 | 18, 17 | brcnv 5896 | . . . . . 6 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
21 | 16, 19, 20 | 3bitri 297 | . . . . 5 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏) |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂))) → (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏)) |
23 | 5, 6, 10, 11, 22 | pospropd 18385 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ((ODual‘𝐷) ∈ Poset ↔ 𝑂 ∈ Poset)) |
24 | 4, 23 | imbitrid 244 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝐷 ∈ Poset → 𝑂 ∈ Poset)) |
25 | 2, 24 | impbid2 226 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ◡ccnv 5688 ‘cfv 6563 Basecbs 17245 lecple 17305 ODualcodu 18343 Posetcpo 18365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ple 17318 df-odu 18344 df-proset 18352 df-poset 18371 |
This theorem is referenced by: odulatb 18492 oduclatb 18565 |
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