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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 2738 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | posglbdg.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | oduleval 18138 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) |
4 | posglbdg.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
5 | eqid 2738 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 1, 5 | odubas 18140 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
7 | 4, 6 | eqtrdi 2794 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) |
8 | posglbdg.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
9 | posglbdg.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | eqid 2738 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
11 | 1, 10 | odulub 18256 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
13 | 8, 12 | eqtrd 2778 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) |
14 | 1 | odupos 18177 | . . 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 3448 | . . . . 5 ⊢ 𝑥 ∈ V | |
20 | brcnvg 5834 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | |
21 | 19, 17, 20 | sylancr 588 | . . . 4 ⊢ (𝜑 → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
22 | 21 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
23 | 18, 22 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥◡ ≤ 𝑇) |
24 | vex 3448 | . . . . . 6 ⊢ 𝑦 ∈ V | |
25 | 19, 24 | brcnv 5837 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) |
26 | 25 | ralbii 3095 | . . . 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 5834 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
30 | 17, 24, 29 | sylancl 587 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
31 | 30 | 3ad2ant1 1134 | . . 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 18263 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3063 Vcvv 3444 ⊆ wss 3909 class class class wbr 5104 ◡ccnv 5631 ‘cfv 6494 Basecbs 17043 lecple 17100 ODualcodu 18135 Posetcpo 18156 lubclub 18158 glbcglb 18159 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-dec 12578 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ple 17113 df-odu 18136 df-proset 18144 df-poset 18162 df-lub 18195 df-glb 18196 |
This theorem is referenced by: mrelatglb 18409 mrelatglb0 18410 glbsscl 46889 ipoglb 46911 |
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