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Theorem vcrel 29813
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 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vc 29812 . 2 CVecOLD = {βŸ¨π‘”, π‘ βŸ© ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ∧ βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))}
21relopabiv 5821 1 Rel CVecOLD
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   Γ— cxp 5675  ran crn 5678  Rel wrel 5682  βŸΆwf 6540  (class class class)co 7409  β„‚cc 11108  1c1 11111   + caddc 11113   Β· cmul 11115  AbelOpcablo 29797  CVecOLDcvc 29811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-vc 29812
This theorem is referenced by:  vcex  29831  nvvop  29862  phop  30071
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