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Mirrors > Home > HSE Home > Th. List > ch0le | Structured version Visualization version GIF version |
Description: The zero subspace is the smallest member of Cℋ. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le | ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 31253 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | sh0le 31469 | . 2 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3963 Sℋ csh 30957 Cℋ cch 30958 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-uni 4913 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-iota 6516 df-fv 6571 df-ov 7434 df-sh 31236 df-ch 31250 df-ch0 31282 |
This theorem is referenced by: chnlen0 31473 ch0pss 31474 ch0lei 31480 chssoc 31525 atcveq0 32377 |
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