Theorem vcabl 8172

Description: Vector addition is an Abelian group operation.

Hypothesis

Ref	Expression
vcabl.1	⊢ G = (1st ‘W)

Assertion

Ref	Expression
vcabl	⊢ (W ∈ CVec → G ∈ Abel)

Proof of Theorem vcabl

Step	Hyp	Ref	Expression
1		vcabl.1	. . 3 ⊢ G = (1st ‘W)
2		eqid 1478	. . 3 ⊢ (2nd ‘W) = (2nd ‘W)
3		eqid 1478	. . 3 ⊢ ran G = ran G
4	1, 2, 3	vci 8163	. 2 ⊢ (W ∈ CVec → (G ∈ Abel ⋀ (2nd ‘W):(ℂ × ran G)–→ran G ⋀ ∀x ∈ ran G((1(2nd ‘W)x) = x ⋀ ∀y ∈ ℂ (∀z ∈ ran G(y(2nd ‘W)(xGz)) = ((y(2nd ‘W)x)G(y(2nd ‘W)z)) ⋀ ∀z ∈ ℂ (((y + z)(2nd ‘W)x) = ((y(2nd ‘W)x)G(z(2nd ‘W)x)) ⋀ ((y · z)(2nd ‘W)x) = (y(2nd ‘W)(z(2nd ‘W)x)))))))
5	4	3simp1d 796	1 ⊢ (W ∈ CVec → G ∈ Abel)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648   × cxp 3174  ran crn 3177  –→wf 3184   ‘cfv 3188  (class class class)co 3969  1^st c1st 4083  2^nd c2nd 4084  ℂcc 5244  1c1 5247   + caddc 5249   · cmul 5251  Abelcabl 8095  CVeccvc 8160

This theorem is referenced by:  vcgrp 8173  vccom 8175  vca23 8177  vca4 8178  nvabl 8231  ip0i 8480  ipdirilem 8484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-vc 8161

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