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Theorem nvss 30622
Description: Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvss NrmCVec ⊆ (CVecOLD × V)

Proof of Theorem nvss
Dummy variables 𝑔 𝑠 𝑛 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2827 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑠⟩ → (𝑤 ∈ CVecOLD ↔ ⟨𝑔, 𝑠⟩ ∈ CVecOLD))
21biimpar 477 . . . . . 6 ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ ⟨𝑔, 𝑠⟩ ∈ CVecOLD) → 𝑤 ∈ CVecOLD)
323ad2antr1 1187 . . . . 5 ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))) → 𝑤 ∈ CVecOLD)
43exlimivv 1930 . . . 4 (∃𝑔𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))) → 𝑤 ∈ CVecOLD)
5 vex 3482 . . . 4 𝑛 ∈ V
64, 5jctir 520 . . 3 (∃𝑔𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))) → (𝑤 ∈ CVecOLD𝑛 ∈ V))
76ssopab2i 5560 . 2 {⟨𝑤, 𝑛⟩ ∣ ∃𝑔𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦)))))} ⊆ {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD𝑛 ∈ V)}
8 df-nv 30621 . . 3 NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))}
9 dfoprab2 7491 . . 3 {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))} = {⟨𝑤, 𝑛⟩ ∣ ∃𝑔𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦)))))}
108, 9eqtri 2763 . 2 NrmCVec = {⟨𝑤, 𝑛⟩ ∣ ∃𝑔𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦)))))}
11 df-xp 5695 . 2 (CVecOLD × V) = {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD𝑛 ∈ V)}
127, 10, 113sstr4i 4039 1 NrmCVec ⊆ (CVecOLD × V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  {coprab 7432  cc 11151  cr 11152  0cc0 11153   + caddc 11156   · cmul 11158  cle 11294  abscabs 15270  GIdcgi 30519  CVecOLDcvc 30587  NrmCVeccnv 30613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-oprab 7435  df-nv 30621
This theorem is referenced by:  nvvcop  30623  nvrel  30631  nvvop  30638  nvex  30640
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