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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 2610 | . . 3 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | posglbd.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | oduleval 16954 | . 2 ⊢ ◡ ≤ = (le‘(ODual‘𝐾)) |
4 | posglbd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
5 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 1, 5 | odubas 16956 | . . 3 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
7 | 4, 6 | syl6eq 2660 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘(ODual‘𝐾))) |
8 | posglbd.g | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | |
9 | posglbd.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
10 | eqid 2610 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
11 | 1, 10 | odulub 16964 | . . . 4 ⊢ (𝐾 ∈ Poset → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (glb‘𝐾) = (lub‘(ODual‘𝐾))) |
13 | 8, 12 | eqtrd 2644 | . 2 ⊢ (𝜑 → 𝐺 = (lub‘(ODual‘𝐾))) |
14 | 1 | odupos 16958 | . . 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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
20 | brcnvg 5225 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑇 ∈ 𝐵) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) | |
21 | 19, 17, 20 | sylancr 694 | . . . 4 ⊢ (𝜑 → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
22 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥◡ ≤ 𝑇 ↔ 𝑇 ≤ 𝑥)) |
23 | 18, 22 | mpbird 246 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥◡ ≤ 𝑇) |
24 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
25 | 19, 24 | brcnv 5227 | . . . . 5 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥) |
26 | 25 | ralbii 2963 | . . . 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 1356 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑦 ≤ 𝑇) |
29 | brcnvg 5225 | . . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) | |
30 | 17, 24, 29 | sylancl 693 | . . . 4 ⊢ (𝜑 → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
31 | 30 | 3ad2ant1 1075 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → (𝑇◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑇)) |
32 | 28, 31 | mpbird 246 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥◡ ≤ 𝑦) → 𝑇◡ ≤ 𝑦) |
33 | 3, 7, 13, 15, 16, 17, 23, 32 | poslubdg 16972 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ◡ccnv 5037 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 lubclub 16765 glbcglb 16766 ODualcodu 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ple 15788 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-odu 16952 |
This theorem is referenced by: mrelatglb 17007 mrelatglb0 17008 |
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