Theorem nvss 30886

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.

Step	Hyp	Ref	Expression
1		eleq1 2857	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑠⟩ → (𝑤 ∈ CVecOLD ↔ ⟨𝑔, 𝑠⟩ ∈ CVecOLD))
2	1	biimpar 482	. . . . . 6 ⊢ ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ ⟨𝑔, 𝑠⟩ ∈ CVecOLD) → 𝑤 ∈ CVecOLD)
3	2	3ad2antr1 1205	. . . . 5 ⊢ ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))) → 𝑤 ∈ CVecOLD)
4	3	exlimivv 1959	. . . 4 ⊢ (∃𝑔∃𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))) → 𝑤 ∈ CVecOLD)
5		vex 3467	. . . 4 ⊢ 𝑛 ∈ V
6	4, 5	jctir 529	. . 3 ⊢ (∃𝑔∃𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))) → (𝑤 ∈ CVecOLD ∧ 𝑛 ∈ V))
7	6	ssopab2i 5536	. 2 ⊢ {⟨𝑤, 𝑛⟩ ∣ ∃𝑔∃𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))} ⊆ {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD ∧ 𝑛 ∈ V)}
8		df-nv 30885	. . 3 ⊢ NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))}
9		dfoprab2 7469	. . 3 ⊢ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))} = {⟨𝑤, 𝑛⟩ ∣ ∃𝑔∃𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))}
10	8, 9	eqtri 2792	. 2 ⊢ NrmCVec = {⟨𝑤, 𝑛⟩ ∣ ∃𝑔∃𝑠(𝑤 = ⟨𝑔, 𝑠⟩ ∧ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))}
11		df-xp 5668	. 2 ⊢ (CVecOLD × V) = {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD ∧ 𝑛 ∈ V)}
12	7, 10, 11	3sstr4i 3996	1 ⊢ NrmCVec ⊆ (CVecOLD × V)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ⟨cop 4600   class class class wbr 5113  {copab 5177   × cxp 5660  ran crn 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  {coprab 7412  ℂcc 11098  ℝcr 11099  0cc0 11100   + caddc 11103   · cmul 11105   ≤ cle 11244  abscabs 15285  GIdcgi 30783  CVec_OLDcvc 30851  NrmCVeccnv 30877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668  df-oprab 7415  df-nv 30885

This theorem is referenced by:  nvvcop  30887  nvrel  30895  nvvop  30902  nvex  30904