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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 2622 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | posglbd.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | oduleval 17131 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) |
4 | posglbd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
5 | eqid 2622 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 1, 5 | odubas 17133 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
7 | 4, 6 | syl6eq 2672 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) |
8 | posglbd.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
9 | posglbd.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | eqid 2622 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
11 | 1, 10 | odulub 17141 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
13 | 8, 12 | eqtrd 2656 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) |
14 | 1 | odupos 17135 | . . 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 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
20 | brcnvg 5303 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | |
21 | 19, 17, 20 | sylancr 695 | . . . 4 ⊢ (𝜑 → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
22 | 21 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
23 | 18, 22 | mpbird 247 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥◡ ≤ 𝑇) |
24 | vex 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
25 | 19, 24 | brcnv 5305 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) |
26 | 25 | ralbii 2980 | . . . 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 1364 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑦 ≤ 𝑇) |
29 | brcnvg 5303 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
30 | 17, 24, 29 | sylancl 694 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
31 | 30 | 3ad2ant1 1082 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
32 | 28, 31 | mpbird 247 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑇◡ ≤ 𝑦) |
33 | 3, 7, 13, 15, 16, 17, 23, 32 | poslubdg 17149 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ◡ccnv 5113 ‘cfv 5888 Basecbs 15857 lecple 15948 Posetcpo 16940 lubclub 16942 glbcglb 16943 ODualcodu 17128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ple 15961 df-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-odu 17129 |
This theorem is referenced by: mrelatglb 17184 mrelatglb0 17185 |
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