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Mirrors > Home > MPE Home > Th. List > posglbd | 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 |
---|---|
posglbd.l | ⊢ ≤ = (le‘𝐾) |
posglbd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
posglbd.g | ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) |
posglbd.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
posglbd.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
posglbd.t | ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
posglbd.lb | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥) |
posglbd.gt | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇) |
Ref | Expression |
---|---|
posglbd | ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | posglbd.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | oduleval 17735 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) |
4 | posglbd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 1, 5 | odubas 17737 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
7 | 4, 6 | syl6eq 2872 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) |
8 | posglbd.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
9 | posglbd.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | eqid 2821 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
11 | 1, 10 | odulub 17745 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
13 | 8, 12 | eqtrd 2856 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) |
14 | 1 | odupos 17739 | . . 3 ⊢ (𝐾 ∈ Poset → (ODual‘𝐾) ∈ Poset) |
15 | 9, 14 | syl 17 | . 2 ⊢ (𝜑 → (ODual‘𝐾) ∈ Poset) |
16 | posglbd.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
17 | posglbd.t | . 2 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
18 | posglbd.lb | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥) | |
19 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
20 | brcnvg 5744 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | |
21 | 19, 17, 20 | sylancr 589 | . . . 4 ⊢ (𝜑 → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
22 | 21 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
23 | 18, 22 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥◡ ≤ 𝑇) |
24 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
25 | 19, 24 | brcnv 5747 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) |
26 | 25 | ralbii 3165 | . . . 4 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦 ↔ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) |
27 | posglbd.gt | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇) | |
28 | 26, 27 | syl3an3b 1401 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑦 ≤ 𝑇) |
29 | brcnvg 5744 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
30 | 17, 24, 29 | sylancl 588 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
31 | 30 | 3ad2ant1 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
32 | 28, 31 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑇◡ ≤ 𝑦) |
33 | 3, 7, 13, 15, 16, 17, 23, 32 | poslubdg 17753 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ◡ccnv 5548 ‘cfv 6349 Basecbs 16477 lecple 16566 Posetcpo 17544 lubclub 17546 glbcglb 17547 ODualcodu 17732 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ple 16579 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-odu 17733 |
This theorem is referenced by: mrelatglb 17788 mrelatglb0 17789 |
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