Theorem vcablo 30658

Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
vcabl.1	⊢ 𝐺 = (1st ‘𝑊)

Assertion

Ref	Expression
vcablo	⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)

Proof of Theorem vcablo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vcabl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2		eqid 2739	. . 3 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊)
3		eqid 2739	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	vciOLD 30650	. 2 ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ (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	simp1d 1148	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   × cxp 5616  ran crn 5619  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11027  1c1 11030   + caddc 11032   · cmul 11034  AbelOpcablo 30633  CVec_OLDcvc 30647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-1st 7931  df-2nd 7932  df-vc 30648

This theorem is referenced by:  vcgrp  30659  nvablo  30705  ip0i  30914  ipdirilem  30918