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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atpsubclN | Structured version Visualization version GIF version |
Description: A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1psubcl.a | ⊢ 𝐴 = (Atoms‘𝐾) |
1psubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
atpsubclN | ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4813 | . . 3 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ⊆ 𝐴) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ⊆ 𝐴) |
3 | 1psubcl.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2735 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
5 | 3, 4 | 2polatN 39915 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}) |
6 | 1psubcl.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 3, 4, 6 | ispsubclN 39920 | . . 3 ⊢ (𝐾 ∈ HL → ({𝑄} ∈ 𝐶 ↔ ({𝑄} ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}))) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ({𝑄} ∈ 𝐶 ↔ ({𝑄} ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}))) |
9 | 2, 5, 8 | mpbir2and 713 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Atomscatm 39245 HLchlt 39332 ⊥𝑃cpolN 39885 PSubClcpscN 39917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-pmap 39487 df-polarityN 39886 df-psubclN 39918 |
This theorem is referenced by: pclfinclN 39933 |
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