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Theorem sspid 28135
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 3848 . . . 4 ( +𝑣𝑈) ⊆ ( +𝑣𝑈)
2 ssid 3848 . . . 4 ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈)
3 ssid 3848 . . . 4 (normCV𝑈) ⊆ (normCV𝑈)
41, 2, 33pm3.2i 1444 . . 3 (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))
54jctr 522 . 2 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))))
6 eqid 2825 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
7 eqid 2825 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
8 eqid 2825 . . 3 (normCV𝑈) = (normCV𝑈)
9 sspid.h . . 3 𝐻 = (SubSp‘𝑈)
106, 6, 7, 7, 8, 8, 9isssp 28134 . 2 (𝑈 ∈ NrmCVec → (𝑈𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈)))))
115, 10mpbird 249 1 (𝑈 ∈ NrmCVec → 𝑈𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wss 3798  cfv 6123  NrmCVeccnv 27994   +𝑣 cpv 27995   ·𝑠OLD cns 27997  normCVcnmcv 28000  SubSpcss 28131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fo 6129  df-fv 6131  df-oprab 6909  df-1st 7428  df-2nd 7429  df-vc 27969  df-nv 28002  df-va 28005  df-sm 28007  df-nmcv 28010  df-ssp 28132
This theorem is referenced by:  hhsssh  28681
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