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Mirrors > Home > HSE Home > Th. List > ocel | Structured version Visualization version GIF version |
Description: Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ocel | ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocval 29063 | . . 3 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0}) | |
2 | 1 | eleq2d 2875 | . 2 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0})) |
3 | oveq1 7142 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥)) | |
4 | 3 | eqeq1d 2800 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0)) |
5 | 4 | ralbidv 3162 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)) |
6 | 5 | elrab 3628 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)) |
7 | 2, 6 | syl6bb 290 | 1 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℋchba 28702 ·ih csp 28705 ⊥cort 28713 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oc 29035 |
This theorem is referenced by: shocel 29065 ocsh 29066 ocorth 29074 ococss 29076 occllem 29086 occl 29087 chocnul 29111 h1deoi 29332 h1dei 29333 hmopidmpji 29935 |
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