Theorem vcablo 30601

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 2740	. . 3 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊)
3		eqid 2740	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	vciOLD 30593	. 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 1537   ∈ wcel 2108  ∀wral 3067   × cxp 5698  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  ℂcc 11182  1c1 11185   + caddc 11187   · cmul 11189  AbelOpcablo 30576  CVec_OLDcvc 30590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-1st 8030  df-2nd 8031  df-vc 30591

This theorem is referenced by:  vcgrp  30602  nvablo  30648  ip0i  30857  ipdirilem  30861