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Theorem sspnv 28430
Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
sspnv.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspnv ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)

Proof of Theorem sspnv
StepHypRef Expression
1 eqid 2818 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2818 . . 3 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2818 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2818 . . 3 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2818 . . 3 (normCV𝑈) = (normCV𝑈)
6 eqid 2818 . . 3 (normCV𝑊) = (normCV𝑊)
7 sspnv.h . . 3 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 28428 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simprbda 499 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wss 3933  cfv 6348  NrmCVeccnv 28288   +𝑣 cpv 28289   ·𝑠OLD cns 28291  normCVcnmcv 28294  SubSpcss 28425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-oprab 7149  df-1st 7678  df-2nd 7679  df-vc 28263  df-nv 28296  df-va 28299  df-sm 28301  df-nmcv 28304  df-ssp 28426
This theorem is referenced by:  sspg  28432  ssps  28434  sspmlem  28436  sspmval  28437  sspz  28439  sspn  28440  sspimsval  28442  bnsscmcl  28572  minvecolem2  28579  hhshsslem1  28971  hhshsslem2  28972
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