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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 31152 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | sh0le 31368 | . 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 2099 ⊆ wss 3947 Sℋ csh 30856 Cℋ cch 30857 0ℋc0h 30863 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-hilex 30927 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-xp 5679 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fv 6552 df-ov 7417 df-sh 31135 df-ch 31149 df-ch0 31181 |
This theorem is referenced by: chnlen0 31372 ch0pss 31373 ch0lei 31379 chssoc 31424 atcveq0 32276 |
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