![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > sh0le | Structured version Visualization version GIF version |
Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sh0le | ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ch0 31282 | . 2 ⊢ 0ℋ = {0ℎ} | |
2 | sh0 31245 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
3 | 2 | snssd 4814 | . 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴) |
4 | 1, 3 | eqsstrid 4044 | 1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 0ℎc0v 30953 Sℋ csh 30957 0ℋc0h 30964 |
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-ext 2706 ax-sep 5302 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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 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-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-sh 31236 df-ch0 31282 |
This theorem is referenced by: ch0le 31470 shle0 31471 orthin 31475 ssjo 31476 shs0i 31478 span0 31571 |
Copyright terms: Public domain | W3C validator |