Theorem vcrel 29565

Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
vcrel	⊢ Rel CVecOLD

Proof of Theorem vcrel

Dummy variables 𝑔 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vc 29564	. 2 ⊢ CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
2	1	relopabiv 5781	1 ⊢ Rel CVecOLD

Syntax hints:   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060   × cxp 5636  ran crn 5639  Rel wrel 5643  ⟶wf 6497  (class class class)co 7362  ℂcc 11058  1c1 11061   + caddc 11063   · cmul 11065  AbelOpcablo 29549  CVec_OLDcvc 29563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-opab 5173  df-xp 5644  df-rel 5645  df-vc 29564

This theorem is referenced by:  vcex  29583  nvvop  29614  phop  29823