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| Mirrors > Home > MPE Home > Th. List > Mathboxes > psubssat | Structured version Visualization version GIF version | ||
| Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.) |
| Ref | Expression |
|---|---|
| atpsub.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| atpsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
| Ref | Expression |
|---|---|
| psubssat | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 3 | atpsub.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | atpsub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
| 5 | 1, 2, 3, 4 | ispsubsp 39747 | . 2 ⊢ (𝐾 ∈ 𝐵 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋)))) |
| 6 | 5 | simprbda 498 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 Atomscatm 39264 PSubSpcpsubsp 39498 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-psubsp 39505 |
| This theorem is referenced by: psubatN 39757 paddidm 39843 paddclN 39844 paddss 39847 pmodlem1 39848 pmod1i 39850 pmodl42N 39853 elpcliN 39895 pclidN 39898 pclbtwnN 39899 pclunN 39900 pclun2N 39901 pclfinN 39902 polssatN 39910 psubclsubN 39942 |
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