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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pcl0N | Structured version Visualization version GIF version |
Description: The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pcl0.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pcl0N | ⊢ (𝐾 ∈ HL → (𝑈‘∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4393 | . . . 4 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | eqid 2728 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | eqid 2728 | . . . . 5 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | pcl0.c | . . . . 5 ⊢ 𝑈 = (PCl‘𝐾) | |
5 | 2, 3, 4 | pclss2polN 39389 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ∅ ⊆ (Atoms‘𝐾)) → (𝑈‘∅) ⊆ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅))) |
6 | 1, 5 | mpan2 690 | . . 3 ⊢ (𝐾 ∈ HL → (𝑈‘∅) ⊆ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅))) |
7 | 3 | 2pol0N 39379 | . . 3 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘∅)) = ∅) |
8 | 6, 7 | sseqtrd 4019 | . 2 ⊢ (𝐾 ∈ HL → (𝑈‘∅) ⊆ ∅) |
9 | ss0 4395 | . 2 ⊢ ((𝑈‘∅) ⊆ ∅ → (𝑈‘∅) = ∅) | |
10 | 8, 9 | syl 17 | 1 ⊢ (𝐾 ∈ HL → (𝑈‘∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ∅c0 4319 ‘cfv 6543 Atomscatm 38730 HLchlt 38817 PClcpclN 39355 ⊥𝑃cpolN 39370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-psubsp 38971 df-pmap 38972 df-pclN 39356 df-polarityN 39371 |
This theorem is referenced by: pcl0bN 39391 pclfinclN 39418 |
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