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Theorem nvss 28674
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 2825 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑠⟩ → (𝑤 ∈ CVecOLD ↔ ⟨𝑔, 𝑠⟩ ∈ CVecOLD))
21biimpar 481 . . . . . 6 ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ ⟨𝑔, 𝑠⟩ ∈ CVecOLD) → 𝑤 ∈ CVecOLD)
323ad2antr1 1190 . . . . 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 1940 . . . 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 3412 . . . 4 𝑛 ∈ V
64, 5jctir 524 . . 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 5431 . 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 28673 . . 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 7269 . . 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 2765 . 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 5557 . 2 (CVecOLD × V) = {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD𝑛 ∈ V)}
127, 10, 113sstr4i 3944 1 NrmCVec ⊆ (CVecOLD × V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  wral 3061  Vcvv 3408  wss 3866  cop 4547   class class class wbr 5053  {copab 5115   × cxp 5549  ran crn 5552  wf 6376  cfv 6380  (class class class)co 7213  {coprab 7214  cc 10727  cr 10728  0cc0 10729   + caddc 10732   · cmul 10734  cle 10868  abscabs 14797  GIdcgi 28571  CVecOLDcvc 28639  NrmCVeccnv 28665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-xp 5557  df-oprab 7217  df-nv 28673
This theorem is referenced by:  nvvcop  28675  nvrel  28683  nvvop  28690  nvex  28692
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