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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 2821 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | psubcli2.p | . . 3 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | psubcli2.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | ispsubclN 37088 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
5 | 4 | simplbda 502 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 Atomscatm 36414 ⊥𝑃cpolN 37053 PSubClcpscN 37085 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-psubclN 37086 |
This theorem is referenced by: psubclsubN 37091 pmapidclN 37093 poml6N 37106 osumcllem3N 37109 osumclN 37118 pmapojoinN 37119 pexmidN 37120 pexmidlem6N 37126 |
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