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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 20012 | . . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )) |
7 | 6 | simp3bi 1078 | . 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) |
8 | oveq2 6658 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵)) | |
9 | 8 | eqeq1d 2624 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 )) |
10 | 9 | rspccva 3308 | . 2 ⊢ ((∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) |
11 | 7, 10 | sylan 488 | 1 ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑖cip 15946 0gc0g 16100 ocvcocv 20004 |
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-8 1992 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 ax-un 6949 |
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-ne 2795 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-ocv 20007 |
This theorem is referenced by: ocvocv 20015 ocvlss 20016 ocvin 20018 lsmcss 20036 clsocv 23049 |
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