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Theorem sspid 27914
 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 3771 . . . 4 ( +𝑣𝑈) ⊆ ( +𝑣𝑈)
2 ssid 3771 . . . 4 ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈)
3 ssid 3771 . . . 4 (normCV𝑈) ⊆ (normCV𝑈)
41, 2, 33pm3.2i 1422 . . 3 (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))
54jctr 508 . 2 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))))
6 eqid 2770 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
7 eqid 2770 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
8 eqid 2770 . . 3 (normCV𝑈) = (normCV𝑈)
9 sspid.h . . 3 𝐻 = (SubSp‘𝑈)
106, 6, 7, 7, 8, 8, 9isssp 27913 . 2 (𝑈 ∈ NrmCVec → (𝑈𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈)))))
115, 10mpbird 247 1 (𝑈 ∈ NrmCVec → 𝑈𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144   ⊆ wss 3721  ‘cfv 6031  NrmCVeccnv 27773   +𝑣 cpv 27774   ·𝑠OLD cns 27776  normCVcnmcv 27779  SubSpcss 27910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-oprab 6796  df-1st 7314  df-2nd 7315  df-vc 27748  df-nv 27781  df-va 27784  df-sm 27786  df-nmcv 27789  df-ssp 27911 This theorem is referenced by:  hhsssh  28460
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