|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 21686 | . . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )) | 
| 7 | 6 | simp3bi 1148 | . 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) | 
| 8 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵)) | |
| 9 | 8 | eqeq1d 2739 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 )) | 
| 10 | 9 | rspccva 3621 | . 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑖cip 17302 0gc0g 17484 ocvcocv 21678 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-ocv 21681 | 
| This theorem is referenced by: ocvocv 21689 ocvlss 21690 ocvin 21692 lsmcss 21710 clsocv 25284 | 
| Copyright terms: Public domain | W3C validator |