Theorem nvss 30396

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 2816	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑠⟩ → (𝑤 ∈ CVecOLD ↔ ⟨𝑔, 𝑠⟩ ∈ CVecOLD))
2	1	biimpar 477	. . . . . 6 ⊢ ((𝑤 = ⟨𝑔, 𝑠⟩ ∧ ⟨𝑔, 𝑠⟩ ∈ CVecOLD) → 𝑤 ∈ CVecOLD)
3	2	3ad2antr1 1186	. . . . 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 1928	. . . 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 3473	. . . 4 ⊢ 𝑛 ∈ V
6	4, 5	jctir 520	. . 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 5546	. 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 30395	. . 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 7472	. . 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 2755	. 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 5678	. 2 ⊢ (CVecOLD × V) = {⟨𝑤, 𝑛⟩ ∣ (𝑤 ∈ CVecOLD ∧ 𝑛 ∈ V)}
12	7, 10, 11	3sstr4i 4021	1 ⊢ NrmCVec ⊆ (CVecOLD × V)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∀wral 3056  Vcvv 3469   ⊆ wss 3944  ⟨cop 4630   class class class wbr 5142  {copab 5204   × cxp 5670  ran crn 5673  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  {coprab 7415  ℂcc 11130  ℝcr 11131  0cc0 11132   + caddc 11135   · cmul 11137   ≤ cle 11273  abscabs 15207  GIdcgi 30293  CVec_OLDcvc 30361  NrmCVeccnv 30387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5205  df-xp 5678  df-oprab 7418  df-nv 30395

This theorem is referenced by:  nvvcop  30397  nvrel  30405  nvvop  30412  nvex  30414