![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ocvss | Structured version Visualization version GIF version |
Description: The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
ocvss.v | ⊢ 𝑉 = (Base‘𝑊) |
ocvss.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
ocvss | ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvss.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2736 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2736 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2736 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | ocvss.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | elocv 21057 | . . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) |
7 | 6 | simp2bi 1146 | . 2 ⊢ (𝑥 ∈ ( ⊥ ‘𝑆) → 𝑥 ∈ 𝑉) |
8 | 7 | ssriv 3946 | 1 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3062 ⊆ wss 3908 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 Scalarcsca 17128 ·𝑖cip 17130 0gc0g 17313 ocvcocv 21049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7356 df-ocv 21052 |
This theorem is referenced by: ocvocv 21060 ocvlss 21061 ocvlsp 21065 ocv1 21068 cssval 21071 cssss 21074 ocvcss 21076 cssincl 21077 csslss 21080 lsmcss 21081 mrccss 21083 pjcss 21107 csscld 24597 clsocv 24598 |
Copyright terms: Public domain | W3C validator |