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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 4570 | . . 3 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ⊆ 𝐴) | |
2 | 1 | adantl 475 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ⊆ 𝐴) |
3 | 1psubcl.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2777 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
5 | 3, 4 | 2polatN 36070 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}) |
6 | 1psubcl.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 3, 4, 6 | ispsubclN 36075 | . . 3 ⊢ (𝐾 ∈ HL → ({𝑄} ∈ 𝐶 ↔ ({𝑄} ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}))) |
8 | 7 | adantr 474 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ({𝑄} ∈ 𝐶 ↔ ({𝑄} ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘{𝑄})) = {𝑄}))) |
9 | 2, 5, 8 | mpbir2and 703 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → {𝑄} ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 {csn 4397 ‘cfv 6135 Atomscatm 35401 HLchlt 35488 ⊥𝑃cpolN 36040 PSubClcpscN 36072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-riotaBAD 35091 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-undef 7681 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-oposet 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-pmap 35642 df-polarityN 36041 df-psubclN 36073 |
This theorem is referenced by: pclfinclN 36088 |
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