MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspid Structured version   Visualization version   GIF version

Theorem sspid 28504
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 3974 . . . 4 ( +𝑣𝑈) ⊆ ( +𝑣𝑈)
2 ssid 3974 . . . 4 ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈)
3 ssid 3974 . . . 4 (normCV𝑈) ⊆ (normCV𝑈)
41, 2, 33pm3.2i 1336 . . 3 (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))
54jctr 528 . 2 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈))))
6 eqid 2824 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
7 eqid 2824 . . 3 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
8 eqid 2824 . . 3 (normCV𝑈) = (normCV𝑈)
9 sspid.h . . 3 𝐻 = (SubSp‘𝑈)
106, 6, 7, 7, 8, 8, 9isssp 28503 . 2 (𝑈 ∈ NrmCVec → (𝑈𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣𝑈) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑈) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑈) ⊆ (normCV𝑈)))))
115, 10mpbird 260 1 (𝑈 ∈ NrmCVec → 𝑈𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wss 3919  cfv 6343  NrmCVeccnv 28363   +𝑣 cpv 28364   ·𝑠OLD cns 28366  normCVcnmcv 28369  SubSpcss 28500
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351  df-oprab 7149  df-1st 7679  df-2nd 7680  df-vc 28338  df-nv 28371  df-va 28374  df-sm 28376  df-nmcv 28379  df-ssp 28501
This theorem is referenced by:  hhsssh  29048
  Copyright terms: Public domain W3C validator