Theorem vcablo 30589

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 2736	. . 3 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊)
3		eqid 2736	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	vciOLD 30581	. 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 1142	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060   × cxp 5682  ran crn 5685  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  ℂcc 11154  1c1 11157   + caddc 11159   · cmul 11161  AbelOpcablo 30564  CVec_OLDcvc 30578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-1st 8015  df-2nd 8016  df-vc 30579

This theorem is referenced by:  vcgrp  30590  nvablo  30636  ip0i  30845  ipdirilem  30849