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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 4000 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ax-1 6 | . . . . 5 ⊢ (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)) | |
3 | 2 | rgen 3059 | . . . 4 ⊢ ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴) |
4 | 3 | rgen2w 3062 | . . 3 ⊢ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴) |
5 | 1, 4 | pm3.2i 470 | . 2 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝐴)) |
6 | eqid 2728 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | eqid 2728 | . . 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 39212 | . 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 258 | 1 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ⊆ wss 3945 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 lecple 17233 joincjn 18296 Atomscatm 38729 PSubSpcpsubsp 38963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-psubsp 38970 |
This theorem is referenced by: pclvalN 39357 pclclN 39358 |
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