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Mirrors > Home > HSE Home > Th. List > chocvali | Structured version Visualization version GIF version |
Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of 𝐴 is the set of vectors that are orthogonal to all vectors in 𝐴. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chocval.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
chocvali | ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chocval.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | chssii 30120 | . 2 ⊢ 𝐴 ⊆ ℋ |
3 | ocval 30169 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 ⊆ wss 3910 ‘cfv 6496 (class class class)co 7356 0cc0 11050 ℋchba 29808 ·ih csp 29811 Cℋ cch 29818 ⊥cort 29819 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-hilex 29888 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7359 df-sh 30096 df-ch 30110 df-oc 30141 |
This theorem is referenced by: (None) |
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