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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubcli2N | Structured version Visualization version GIF version |
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubcli2.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
psubcli2.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
psubcli2N | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | psubcli2.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | psubcli2.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | ispsubclN 39274 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
5 | 4 | simplbda 499 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ‘cfv 6543 Atomscatm 38599 ⊥𝑃cpolN 39239 PSubClcpscN 39271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-psubclN 39272 |
This theorem is referenced by: psubclsubN 39277 pmapidclN 39279 poml6N 39292 osumcllem3N 39295 osumclN 39304 pmapojoinN 39305 pexmidN 39306 pexmidlem6N 39312 |
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