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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 2740 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | pmapsubcl.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 39716 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | eqid 2740 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
6 | 1, 3, 5 | 2polpmapN 39870 | . 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 39894 | . . 3 ⊢ (𝐾 ∈ HL → ((𝑀‘𝑋) ∈ 𝐶 ↔ ((𝑀‘𝑋) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘(𝑀‘𝑋))) = (𝑀‘𝑋)))) |
9 | 8 | adantr 480 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ‘cfv 6573 Basecbs 17258 Atomscatm 39219 HLchlt 39306 pmapcpmap 39454 ⊥𝑃cpolN 39859 PSubClcpscN 39891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-pmap 39461 df-polarityN 39860 df-psubclN 39892 |
This theorem is referenced by: psubclinN 39905 paddatclN 39906 linepsubclN 39908 polsubclN 39909 pmapojoinN 39925 |
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