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 3989 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | eqid 2821 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2821 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
4 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | eqid 2821 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
6 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
7 | 2, 3, 4, 5, 6 | polval2N 37057 | . . 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 36513 | . . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2821 | . . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 4 | atbase 36440 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
12 | eqid 2821 | . . . . . . . . . . 11 ⊢ (le‘𝐾) = (le‘𝐾) | |
13 | eqid 2821 | . . . . . . . . . . 11 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
14 | 10, 12, 13 | ople1 36342 | . . . . . . . . . 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 3182 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾)(1.‘𝐾)) |
17 | rabid2 3381 | . . . . . . . 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 6674 | . . . . . 6 ⊢ (𝐾 ∈ HL → ((lub‘𝐾)‘𝐴) = ((lub‘𝐾)‘{𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)(1.‘𝐾)})) |
20 | hlomcmat 36516 | . . . . . . 7 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
21 | 10, 13 | op1cl 36336 | . . . . . . . 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 36470 | . . . . . . 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 2857 | . . . . 5 ⊢ (𝐾 ∈ HL → (1.‘𝐾) = ((lub‘𝐾)‘𝐴)) |
26 | 25 | fveq2d 6674 | . . . 4 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘(1.‘𝐾)) = ((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) |
27 | eqid 2821 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
28 | 27, 13, 3 | opoc1 36353 | . . . . 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 2858 | . . 3 ⊢ (𝐾 ∈ HL → ((oc‘𝐾)‘((lub‘𝐾)‘𝐴)) = (0.‘𝐾)) |
31 | 30 | fveq2d 6674 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘((oc‘𝐾)‘((lub‘𝐾)‘𝐴))) = ((pmap‘𝐾)‘(0.‘𝐾))) |
32 | hlatl 36511 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
33 | 27, 5 | pmap0 36916 | . . 3 ⊢ (𝐾 ∈ AtLat → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ((pmap‘𝐾)‘(0.‘𝐾)) = ∅) |
35 | 8, 31, 34 | 3eqtrd 2860 | 1 ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 occoc 16573 lubclub 17552 0.cp0 17647 1.cp1 17648 CLatccla 17717 OPcops 36323 OMLcoml 36326 Atomscatm 36414 AtLatcal 36415 HLchlt 36501 pmapcpmap 36648 ⊥𝑃cpolN 37053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-undef 7939 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-pmap 36655 df-polarityN 37054 |
This theorem is referenced by: 2pol0N 37062 1psubclN 37095 |
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