![]() |
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 2735 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | posglbdg.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | oduleval 18346 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) |
4 | posglbdg.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
5 | eqid 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 1, 5 | odubas 18348 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
7 | 4, 6 | eqtrdi 2791 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) |
8 | posglbdg.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
9 | posglbdg.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | eqid 2735 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
11 | 1, 10 | odulub 18465 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
13 | 8, 12 | eqtrd 2775 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) |
14 | 1 | odupos 18386 | . . 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 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
20 | brcnvg 5893 | . . . . 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 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
25 | 19, 24 | brcnv 5896 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) |
26 | 25 | ralbii 3091 | . . . 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 1404 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑦 ≤ 𝑇) |
29 | brcnvg 5893 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
30 | 17, 24, 29 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
31 | 30 | 3ad2ant1 1132 | . . 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 18472 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ◡ccnv 5688 ‘cfv 6563 Basecbs 17245 lecple 17305 ODualcodu 18343 Posetcpo 18365 lubclub 18367 glbcglb 18368 |
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-rep 5285 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-rmo 3378 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 df-lub 18404 df-glb 18405 |
This theorem is referenced by: mrelatglb 18618 mrelatglb0 18619 glbsscl 48758 ipoglb 48780 |
Copyright terms: Public domain | W3C validator |