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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 2798 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | pmapsubcl.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 37055 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | eqid 2798 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
6 | 1, 3, 5 | 2polpmapN 37209 | . 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 37233 | . . 3 ⊢ (𝐾 ∈ HL → ((𝑀‘𝑋) ∈ 𝐶 ↔ ((𝑀‘𝑋) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)))) |
9 | 8 | adantr 484 | . 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 712 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 Basecbs 16475 Atomscatm 36559 HLchlt 36646 pmapcpmap 36793 ⊥𝑃cpolN 37198 PSubClcpscN 37230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-undef 7922 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-pmap 36800 df-polarityN 37199 df-psubclN 37231 |
This theorem is referenced by: psubclinN 37244 paddatclN 37245 linepsubclN 37247 polsubclN 37248 pmapojoinN 37264 |
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