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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 20740 | . . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )) |
7 | 6 | simp3bi 1139 | . 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) |
8 | oveq2 7153 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵)) | |
9 | 8 | eqeq1d 2820 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 )) |
10 | 9 | rspccva 3619 | . 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 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑖cip 16558 0gc0g 16701 ocvcocv 20732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-ocv 20735 |
This theorem is referenced by: ocvocv 20743 ocvlss 20744 ocvin 20746 lsmcss 20764 clsocv 23780 |
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