Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 4733 | . . 3 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ) | |
3 | ocel 29050 | . . 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 3512 | . . . 4 ⊢ 𝐵 ∈ V |
6 | oveq2 7156 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐴 ·ih 𝐵)) | |
7 | 6 | eqeq1d 2821 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0)) |
8 | 5, 7 | ralsn 4611 | . . 3 ⊢ (∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0) |
9 | 8 | anbi2i 624 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) |
10 | 4, 9 | bitri 277 | 1 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ⊆ wss 3934 {csn 4559 ‘cfv 6348 (class class class)co 7148 0cc0 10529 ℋchba 28688 ·ih csp 28691 ⊥cort 28699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-hilex 28768 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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-iota 6307 df-fun 6350 df-fv 6356 df-ov 7151 df-oc 29021 |
This theorem is referenced by: h1dei 29319 |
Copyright terms: Public domain | W3C validator |