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Theorem sspnv 28509
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 2798 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2798 . . 3 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2798 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2798 . . 3 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2798 . . 3 (normCV𝑈) = (normCV𝑈)
6 eqid 2798 . . 3 (normCV𝑊) = (normCV𝑊)
7 sspnv.h . . 3 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 28507 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simprbda 502 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wss 3881  cfv 6324  NrmCVeccnv 28367   +𝑣 cpv 28368   ·𝑠OLD cns 28370  normCVcnmcv 28373  SubSpcss 28504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-oprab 7139  df-1st 7671  df-2nd 7672  df-vc 28342  df-nv 28375  df-va 28378  df-sm 28380  df-nmcv 28383  df-ssp 28505
This theorem is referenced by:  sspg  28511  ssps  28513  sspmlem  28515  sspmval  28516  sspz  28518  sspn  28519  sspimsval  28521  bnsscmcl  28651  minvecolem2  28658  hhshsslem1  29050  hhshsslem2  29051
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