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 3992 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | eqid 2824 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2824 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
4 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | eqid 2824 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
6 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
7 | 2, 3, 4, 5, 6 | polval2N 37046 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝐴 ⊆ 𝐴) → ( ⊥ ‘𝐴) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴)))) |
8 | 1, 7 | mpan2 689 | . 2 ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴)))) |
9 | hlop 36502 | . . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2824 | . . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 4 | atbase 36429 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
12 | eqid 2824 | . . . . . . . . . . 11 ⊢ (le‘𝐾) = (le‘𝐾) | |
13 | eqid 2824 | . . . . . . . . . . 11 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
14 | 10, 12, 13 | ople1 36331 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾)(1.‘𝐾)) |
15 | 9, 11, 14 | syl2an 597 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾)(1.‘𝐾)) |
16 | 15 | ralrimiva 3185 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾)(1.‘𝐾)) |
17 | rabid2 3384 | . . . . . . . 8 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)} ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾)(1.‘𝐾)) | |
18 | 16, 17 | sylibr 236 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)}) |
19 | 18 | fveq2d 6677 | . . . . . 6 ⊢ (𝐾 ∈ HL → ((lub‘𝐾)‘𝐴) = ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)})) |
20 | hlomcmat 36505 | . . . . . . 7 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
21 | 10, 13 | op1cl 36325 | . . . . . . . 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 36459 | . . . . . . 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 586 | . . . . . 6 ⊢ (𝐾 ∈ HL → ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)}) = (1.‘𝐾)) |
25 | 19, 24 | eqtr2d 2860 | . . . . 5 ⊢ (𝐾 ∈ HL → (1.‘𝐾) = ((lub‘𝐾)‘𝐴)) |
26 | 25 | fveq2d 6677 | . . . 4 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘(1.‘𝐾)) = ((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) |
27 | eqid 2824 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
28 | 27, 13, 3 | opoc1 36342 | . . . . 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 2861 | . . 3 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘((lub‘𝐾)‘𝐴)) = (0.‘𝐾)) |
31 | 30 | fveq2d 6677 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) = ((pmap‘𝐾)‘(0.‘𝐾))) |
32 | hlatl 36500 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
33 | 27, 5 | pmap0 36905 | . . 3 ⊢ (𝐾 ∈ AtLat → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
35 | 8, 31, 34 | 3eqtrd 2863 | 1 ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 {crab 3145 ⊆ wss 3939 ∅c0 4294 class class class wbr 5069 ‘cfv 6358 Basecbs 16486 lecple 16575 occoc 16576 lubclub 17555 0.cp0 17650 1.cp1 17651 CLatccla 17720 OPcops 36312 OMLcoml 36315 Atomscatm 36403 AtLatcal 36404 HLchlt 36490 pmapcpmap 36637 ⊥𝑃cpolN 37042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-undef 7942 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-pmap 36644 df-polarityN 37043 |
This theorem is referenced by: 2pol0N 37051 1psubclN 37084 |
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