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Mirrors > Home > HSE Home > Th. List > atcv0 | Structured version Visualization version GIF version |
Description: An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atcv0 | ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ela 31455 | . 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5141 Cℋ cch 30045 0ℋc0h 30051 ⋖ℋ ccv 30080 HAtomscat 30081 |
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-ext 2702 |
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-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-at 31454 |
This theorem is referenced by: atcveq0 31464 atcv0eq 31495 |
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