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Mirrors > Home > MPE Home > Th. List > ocvi | Structured version Visualization version GIF version |
Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
ocvfval.v | ⊢ 𝑉 = (Base‘𝑊) |
ocvfval.i | ⊢ , = (·𝑖‘𝑊) |
ocvfval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
ocvfval.z | ⊢ 0 = (0g‘𝐹) |
ocvfval.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
ocvi | ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvfval.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | ocvfval.i | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
3 | ocvfval.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | ocvfval.z | . . . 4 ⊢ 0 = (0g‘𝐹) | |
5 | ocvfval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | elocv 21212 | . . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )) |
7 | 6 | simp3bi 1147 | . 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) |
8 | oveq2 7413 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵)) | |
9 | 8 | eqeq1d 2734 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 )) |
10 | 9 | rspccva 3611 | . 2 ⊢ ((∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) |
11 | 7, 10 | sylan 580 | 1 ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 ·𝑖cip 17198 0gc0g 17381 ocvcocv 21204 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-ocv 21207 |
This theorem is referenced by: ocvocv 21215 ocvlss 21216 ocvin 21218 lsmcss 21236 clsocv 24758 |
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