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Theorem nvss 28061
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 2870 . . . . . . 7 (𝑤 = ⟨𝑔, 𝑠⟩ → (𝑤 ∈ CVecOLD ↔ ⟨𝑔, 𝑠⟩ ∈ CVecOLD))
21biimpar 478 . . . . . 6 ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ ⟨𝑔, 𝑠⟩ ∈ CVecOLD) → 𝑤 ∈ CVecOLD)
323ad2antr1 1181 . . . . 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 1910 . . . 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 3440 . . . 4 𝑛 ∈ V
64, 5jctir 521 . . 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 5325 . 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 28060 . . 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 7071 . . 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 2819 . 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 5449 . 2 (CVecOLD × V) = {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD𝑛 ∈ V)}
127, 10, 113sstr4i 3931 1 NrmCVec ⊆ (CVecOLD × V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  wral 3105  Vcvv 3437  wss 3859  cop 4478   class class class wbr 4962  {copab 5024   × cxp 5441  ran crn 5444  wf 6221  cfv 6225  (class class class)co 7016  {coprab 7017  cc 10381  cr 10382  0cc0 10383   + caddc 10386   · cmul 10388  cle 10522  abscabs 14427  GIdcgi 27958  CVecOLDcvc 28026  NrmCVeccnv 28052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-opab 5025  df-xp 5449  df-oprab 7020  df-nv 28060
This theorem is referenced by:  nvvcop  28062  nvrel  28070  nvvop  28077  nvex  28079
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