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Theorem vcablo 28464
 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 2758 . . 3 (2nd𝑊) = (2nd𝑊)
3 eqid 2758 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3vciOLD 28456 . 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 1139 1 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   × cxp 5526  ran crn 5529  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  ℂcc 10586  1c1 10589   + caddc 10591   · cmul 10593  AbelOpcablo 28439  CVecOLDcvc 28453 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-1st 7699  df-2nd 7700  df-vc 28454 This theorem is referenced by:  vcgrp  28465  nvablo  28511  ip0i  28720  ipdirilem  28724
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