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 3972 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | ral0 4076 | . . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅) | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)) |
4 | eqid 2622 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2622 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | eqid 2622 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
7 | 0psub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
8 | 4, 5, 6, 7 | ispsubsp 35031 | . 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)))) |
9 | 3, 8 | mpbiri 248 | 1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 lecple 15948 joincjn 16944 Atomscatm 34550 PSubSpcpsubsp 34782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-psubsp 34789 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |