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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 2736 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | pmapsubcl.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 37815 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | eqid 2736 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
6 | 1, 3, 5 | 2polpmapN 37969 | . 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 37993 | . . 3 ⊢ (𝐾 ∈ HL → ((𝑀‘𝑋) ∈ 𝐶 ↔ ((𝑀‘𝑋) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)))) |
9 | 8 | adantr 482 | . 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 711 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ⊆ wss 3892 ‘cfv 6458 Basecbs 16957 Atomscatm 37319 HLchlt 37406 pmapcpmap 37553 ⊥𝑃cpolN 37958 PSubClcpscN 37990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-riotaBAD 37009 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-undef 8120 df-proset 18058 df-poset 18076 df-plt 18093 df-lub 18109 df-glb 18110 df-join 18111 df-meet 18112 df-p0 18188 df-p1 18189 df-lat 18195 df-clat 18262 df-oposet 37232 df-ol 37234 df-oml 37235 df-covers 37322 df-ats 37323 df-atl 37354 df-cvlat 37378 df-hlat 37407 df-pmap 37560 df-polarityN 37959 df-psubclN 37991 |
This theorem is referenced by: psubclinN 38004 paddatclN 38005 linepsubclN 38007 polsubclN 38008 pmapojoinN 38024 |
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