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Theorem vcablo 27312
 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 2621 . . 3 (2nd𝑊) = (2nd𝑊)
3 eqid 2621 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3vciOLD 27304 . 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 1071 1 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908   × cxp 5082  ran crn 5085  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  ℂcc 9894  1c1 9897   + caddc 9899   · cmul 9901  AbelOpcablo 27286  CVecOLDcvc 27301 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-1st 7128  df-2nd 7129  df-vc 27302 This theorem is referenced by:  vcgrp  27313  nvablo  27359  ip0i  27568  ipdirilem  27572
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