|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > posglbdg | Structured version Visualization version GIF version | ||
| Description: Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| posglbdg.l | ⊢ ≤ = (le‘𝐾) | 
| posglbdg.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | 
| posglbdg.g | ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | 
| posglbdg.k | ⊢ (𝜑 → 𝐾 ∈ Poset) | 
| posglbdg.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | 
| posglbdg.t | ⊢ (𝜑 → 𝑇 ∈ 𝐵) | 
| posglbdg.lb | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥) | 
| posglbdg.gt | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇) | 
| Ref | Expression | 
|---|---|
| posglbdg | ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
| 2 | posglbdg.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | 1, 2 | oduleval 18335 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) | 
| 4 | posglbdg.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
| 5 | eqid 2736 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 6 | 1, 5 | odubas 18337 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) | 
| 7 | 4, 6 | eqtrdi 2792 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) | 
| 8 | posglbdg.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
| 9 | posglbdg.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
| 10 | eqid 2736 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 11 | 1, 10 | odulub 18453 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) | 
| 12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) | 
| 13 | 8, 12 | eqtrd 2776 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) | 
| 14 | 1 | odupos 18374 | . . 3 ⊢ (𝐾 ∈ Poset → (ODual‘𝐾) ∈ Poset) | 
| 15 | 9, 14 | syl 17 | . 2 ⊢ (𝜑 → (ODual‘𝐾) ∈ Poset) | 
| 16 | posglbdg.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
| 17 | posglbdg.t | . 2 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
| 18 | posglbdg.lb | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥) | |
| 19 | vex 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 20 | brcnvg 5889 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | |
| 21 | 19, 17, 20 | sylancr 587 | . . . 4 ⊢ (𝜑 → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | 
| 22 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | 
| 23 | 18, 22 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥◡ ≤ 𝑇) | 
| 24 | vex 3483 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 25 | 19, 24 | brcnv 5892 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) | 
| 26 | 25 | ralbii 3092 | . . . 4 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦 ↔ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) | 
| 27 | posglbdg.gt | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇) | |
| 28 | 26, 27 | syl3an3b 1406 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑦 ≤ 𝑇) | 
| 29 | brcnvg 5889 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
| 30 | 17, 24, 29 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | 
| 31 | 30 | 3ad2ant1 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | 
| 32 | 28, 31 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑇◡ ≤ 𝑦) | 
| 33 | 3, 7, 13, 15, 16, 17, 23, 32 | poslubdg 18460 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 class class class wbr 5142 ◡ccnv 5683 ‘cfv 6560 Basecbs 17248 lecple 17305 ODualcodu 18332 Posetcpo 18354 lubclub 18356 glbcglb 18357 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-dec 12736 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ple 17318 df-odu 18333 df-proset 18341 df-poset 18360 df-lub 18392 df-glb 18393 | 
| This theorem is referenced by: mrelatglb 18606 mrelatglb0 18607 glbsscl 48865 ipoglb 48895 | 
| Copyright terms: Public domain | W3C validator |