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Theorem sspid 29028
Description: A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
sspid.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspid (𝑈 ∈ NrmCVec → 𝑈𝐻)

Proof of Theorem sspid
StepHypRef Expression
1 ssid 3944 . . . 4 ( +𝑣𝑈) ⊆ ( +𝑣𝑈)
2 ssid 3944 . . . 4 ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈)
3 ssid 3944 . . . 4 (normCV𝑈) ⊆ (normCV𝑈)
41, 2, 33pm3.2i 1337 . . 3 (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))
54jctr 524 . 2 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))))
6 eqid 2737 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
7 eqid 2737 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
8 eqid 2737 . . 3 (normCV𝑈) = (normCV𝑈)
9 sspid.h . . 3 𝐻 = (SubSp‘𝑈)
106, 6, 7, 7, 8, 8, 9isssp 29027 . 2 (𝑈 ∈ NrmCVec → (𝑈𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈)))))
115, 10mpbird 256 1 (𝑈 ∈ NrmCVec → 𝑈𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1539  wcel 2107  wss 3888  cfv 6423  NrmCVeccnv 28887   +𝑣 cpv 28888   ·𝑠OLD cns 28890  normCVcnmcv 28893  SubSpcss 29024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7571
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3429  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6381  df-fun 6425  df-fn 6426  df-f 6427  df-fo 6429  df-fv 6431  df-oprab 7264  df-1st 7809  df-2nd 7810  df-vc 28862  df-nv 28895  df-va 28898  df-sm 28900  df-nmcv 28903  df-ssp 29025
This theorem is referenced by:  hhsssh  29572
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