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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 3940 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ax-1 6 | . . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)) | |
3 | 2 | rgen 3074 | . . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴) |
4 | 3 | rgen2w 3077 | . . 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 2739 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | eqid 2739 | . . 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 37533 | . 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 1543 ∈ wcel 2112 ∀wral 3064 ⊆ wss 3883 class class class wbr 5070 ‘cfv 6401 (class class class)co 7235 lecple 16842 joincjn 17851 Atomscatm 37051 PSubSpcpsubsp 37284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-iota 6359 df-fun 6403 df-fv 6409 df-ov 7238 df-psubsp 37291 |
This theorem is referenced by: pclvalN 37678 pclclN 37679 |
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