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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 28770 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | sh0le 28988 | . 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 2048 ⊆ wss 3825 Sℋ csh 28474 Cℋ cch 28475 0ℋc0h 28481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 ax-sep 5054 ax-hilex 28545 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-xp 5406 df-cnv 5408 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fv 6190 df-ov 6973 df-sh 28753 df-ch 28767 df-ch0 28799 |
This theorem is referenced by: chnlen0 28992 ch0pss 28993 ch0lei 28999 chssoc 29044 atcveq0 29896 |
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