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Theorem vcablo 26808
 Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
vcabl.1 𝐺 = (1st𝑊)
Assertion
Ref Expression
vcablo (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)

Proof of Theorem vcablo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vcabl.1 . . 3 𝐺 = (1st𝑊)
2 eqid 2610 . . 3 (2nd𝑊) = (2nd𝑊)
3 eqid 2610 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3vciOLD 26800 . 2 (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ (2nd𝑊):(ℂ × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺((1(2nd𝑊)𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝐺(𝑦(2nd𝑊)(𝑥𝐺𝑧)) = ((𝑦(2nd𝑊)𝑥)𝐺(𝑦(2nd𝑊)𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)(2nd𝑊)𝑥) = ((𝑦(2nd𝑊)𝑥)𝐺(𝑧(2nd𝑊)𝑥)) ∧ ((𝑦 · 𝑧)(2nd𝑊)𝑥) = (𝑦(2nd𝑊)(𝑧(2nd𝑊)𝑥)))))))
54simp1d 1066 1 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℂcc 9813  1c1 9816   + caddc 9818   · cmul 9820  AbelOpcablo 26782  CVecOLDcvc 26797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-vc 26798 This theorem is referenced by:  vcgrp  26809  nvablo  26855  ip0i  27064  ipdirilem  27068
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