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Mirrors > Home > HSE Home > Th. List > h1deoi | Structured version Visualization version GIF version |
Description: Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h1deot.1 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
h1deoi | ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h1deot.1 | . . 3 ⊢ 𝐵 ∈ ℋ | |
2 | snssi 4813 | . . 3 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ) | |
3 | ocel 31310 | . . 3 ⊢ ({𝐵} ⊆ ℋ → (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0)) |
5 | 1 | elexi 3501 | . . . 4 ⊢ 𝐵 ∈ V |
6 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐴 ·ih 𝐵)) | |
7 | 6 | eqeq1d 2737 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0)) |
8 | 5, 7 | ralsn 4686 | . . 3 ⊢ (∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0) |
9 | 8 | anbi2i 623 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) |
10 | 4, 9 | bitri 275 | 1 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 {csn 4631 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℋchba 30948 ·ih csp 30951 ⊥cort 30959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oc 31281 |
This theorem is referenced by: h1dei 31579 |
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