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Mirrors > Home > MPE Home > Th. List > Mathboxes > atpsubN | Structured version Visualization version GIF version |
Description: The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atpsub.a | ⊢ 𝐴 = (Atoms‘𝐾) |
atpsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
atpsubN | ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3937 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ax-1 6 | . . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)) | |
3 | 2 | rgen 3116 | . . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴) |
4 | 3 | rgen2w 3119 | . . 3 ⊢ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴) |
5 | 1, 4 | pm3.2i 474 | . 2 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)) |
6 | eqid 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | eqid 2798 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | atpsub.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | atpsub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
10 | 6, 7, 8, 9 | ispsubsp 37041 | . 2 ⊢ (𝐾 ∈ 𝑉 → (𝐴 ∈ 𝑆 ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)))) |
11 | 5, 10 | mpbiri 261 | 1 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 lecple 16564 joincjn 17546 Atomscatm 36559 PSubSpcpsubsp 36792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-psubsp 36799 |
This theorem is referenced by: pclvalN 37186 pclclN 37187 |
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