Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pol1N | Structured version Visualization version GIF version |
Description: The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polssat.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
pol1N | ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3587 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | eqid 2610 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2610 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
4 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | eqid 2610 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
6 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
7 | 2, 3, 4, 5, 6 | polval2N 34210 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝐴 ⊆ 𝐴) → ( ⊥ ‘𝐴) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴)))) |
8 | 1, 7 | mpan2 703 | . 2 ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴)))) |
9 | hlop 33667 | . . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2610 | . . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 4 | atbase 33594 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
12 | eqid 2610 | . . . . . . . . . . 11 ⊢ (le‘𝐾) = (le‘𝐾) | |
13 | eqid 2610 | . . . . . . . . . . 11 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
14 | 10, 12, 13 | ople1 33496 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾)(1.‘𝐾)) |
15 | 9, 11, 14 | syl2an 493 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾)(1.‘𝐾)) |
16 | 15 | ralrimiva 2949 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾)(1.‘𝐾)) |
17 | rabid2 3096 | . . . . . . . 8 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)} ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾)(1.‘𝐾)) | |
18 | 16, 17 | sylibr 223 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)}) |
19 | 18 | fveq2d 6107 | . . . . . 6 ⊢ (𝐾 ∈ HL → ((lub‘𝐾)‘𝐴) = ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)})) |
20 | hlomcmat 33669 | . . . . . . 7 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
21 | 10, 13 | op1cl 33490 | . . . . . . . 8 ⊢ (𝐾 ∈ OP → (1.‘𝐾) ∈ (Base‘𝐾)) |
22 | 9, 21 | syl 17 | . . . . . . 7 ⊢ (𝐾 ∈ HL → (1.‘𝐾) ∈ (Base‘𝐾)) |
23 | 10, 12, 2, 4 | atlatmstc 33624 | . . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (1.‘𝐾) ∈ (Base‘𝐾)) → ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)}) = (1.‘𝐾)) |
24 | 20, 22, 23 | syl2anc 691 | . . . . . 6 ⊢ (𝐾 ∈ HL → ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)}) = (1.‘𝐾)) |
25 | 19, 24 | eqtr2d 2645 | . . . . 5 ⊢ (𝐾 ∈ HL → (1.‘𝐾) = ((lub‘𝐾)‘𝐴)) |
26 | 25 | fveq2d 6107 | . . . 4 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘(1.‘𝐾)) = ((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) |
27 | eqid 2610 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
28 | 27, 13, 3 | opoc1 33507 | . . . . 5 ⊢ (𝐾 ∈ OP → ((oc‘𝐾)‘(1.‘𝐾)) = (0.‘𝐾)) |
29 | 9, 28 | syl 17 | . . . 4 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘(1.‘𝐾)) = (0.‘𝐾)) |
30 | 26, 29 | eqtr3d 2646 | . . 3 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘((lub‘𝐾)‘𝐴)) = (0.‘𝐾)) |
31 | 30 | fveq2d 6107 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) = ((pmap‘𝐾)‘(0.‘𝐾))) |
32 | hlatl 33665 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
33 | 27, 5 | pmap0 34069 | . . 3 ⊢ (𝐾 ∈ AtLat → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
35 | 8, 31, 34 | 3eqtrd 2648 | 1 ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 occoc 15776 lubclub 16765 0.cp0 16860 1.cp1 16861 CLatccla 16930 OPcops 33477 OMLcoml 33480 Atomscatm 33568 AtLatcal 33569 HLchlt 33655 pmapcpmap 33801 ⊥𝑃cpolN 34206 |
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-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-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-undef 7286 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-pmap 33808 df-polarityN 34207 |
This theorem is referenced by: 2pol0N 34215 1psubclN 34248 |
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