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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 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2821 | . . 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 36875 | . 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 501 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 lecple 16566 joincjn 17548 Atomscatm 36393 PSubSpcpsubsp 36626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-psubsp 36633 |
This theorem is referenced by: psubatN 36885 paddidm 36971 paddclN 36972 paddss 36975 pmodlem1 36976 pmod1i 36978 pmodl42N 36981 elpcliN 37023 pclidN 37026 pclbtwnN 37027 pclunN 37028 pclun2N 37029 pclfinN 37030 polssatN 37038 psubclsubN 37070 |
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