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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 2734 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2734 | . . 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 39727 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 lecple 17304 joincjn 18368 Atomscatm 39244 PSubSpcpsubsp 39478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-psubsp 39485 |
This theorem is referenced by: psubatN 39737 paddidm 39823 paddclN 39824 paddss 39827 pmodlem1 39828 pmod1i 39830 pmodl42N 39833 elpcliN 39875 pclidN 39878 pclbtwnN 39879 pclunN 39880 pclun2N 39881 pclfinN 39882 polssatN 39890 psubclsubN 39922 |
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