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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapsubclN | Structured version Visualization version GIF version |
Description: A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmapsubcl.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapsubcl.m | ⊢ 𝑀 = (pmap‘𝐾) |
pmapsubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
pmapsubclN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapsubcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2825 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | pmapsubcl.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 35829 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | eqid 2825 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
6 | 1, 3, 5 | 2polpmapN 35983 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)) |
7 | pmapsubcl.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
8 | 2, 5, 7 | ispsubclN 36007 | . . 3 ⊢ (𝐾 ∈ HL → ((𝑀‘𝑋) ∈ 𝐶 ↔ ((𝑀‘𝑋) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)))) |
9 | 8 | adantr 474 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝐶 ↔ ((𝑀‘𝑋) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)))) |
10 | 4, 6, 9 | mpbir2and 704 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 Basecbs 16229 Atomscatm 35333 HLchlt 35420 pmapcpmap 35567 ⊥𝑃cpolN 35972 PSubClcpscN 36004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-riotaBAD 35023 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-undef 7669 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-pmap 35574 df-polarityN 35973 df-psubclN 36005 |
This theorem is referenced by: psubclinN 36018 paddatclN 36019 linepsubclN 36021 polsubclN 36022 pmapojoinN 36038 |
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