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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 4336 | . . 3 ⊢ ∅ ⊆ (Atoms‘𝐾) | |
2 | ral0 4442 | . . 3 ⊢ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅) | |
3 | 1, 2 | pm3.2i 473 | . 2 ⊢ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)) |
4 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | eqid 2821 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
7 | 0psub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
8 | 4, 5, 6, 7 | ispsubsp 36913 | . 2 ⊢ (𝐾 ∈ 𝑉 → (∅ ∈ 𝑆 ↔ (∅ ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ ∅ ∀𝑞 ∈ ∅ ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∅)))) |
9 | 3, 8 | mpbiri 260 | 1 ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3924 ∅c0 4279 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 lecple 16555 joincjn 17537 Atomscatm 36431 PSubSpcpsubsp 36664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-psubsp 36671 |
This theorem is referenced by: (None) |
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