Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 0psubN | Structured version Visualization version GIF version |
Description: The empty set 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 |
---|---|
0psub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
0psubN | ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4316 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | ral0 4429 | . . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅) | |
3 | 1, 2 | pm3.2i 474 | . 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)) |
4 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | eqid 2737 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
7 | 0psub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
8 | 4, 5, 6, 7 | ispsubsp 37501 | . 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)))) |
9 | 3, 8 | mpbiri 261 | 1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3871 ∅c0 4242 class class class wbr 5058 ‘cfv 6385 (class class class)co 7218 lecple 16814 joincjn 17823 Atomscatm 37019 PSubSpcpsubsp 37252 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-iota 6343 df-fun 6387 df-fv 6393 df-ov 7221 df-psubsp 37259 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |