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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 2733 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2733 | . . 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 38522 | . 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 500 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3946 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 lecple 17191 joincjn 18251 Atomscatm 38039 PSubSpcpsubsp 38273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-psubsp 38280 |
This theorem is referenced by: psubatN 38532 paddidm 38618 paddclN 38619 paddss 38622 pmodlem1 38623 pmod1i 38625 pmodl42N 38628 elpcliN 38670 pclidN 38673 pclbtwnN 38674 pclunN 38675 pclun2N 38676 pclfinN 38677 polssatN 38685 psubclsubN 38717 |
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