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Theorem vcablo 30087

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

**Hypothesis**
Ref	Expression
vcabl.1	⊢ 𝐺 = (1^st ‘𝑊)

**Assertion**
Ref	Expression
vcablo	⊢ (𝑊 ∈ CVec_OLD → 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 = (1^st ‘𝑊)
2		eqid 2730	. . 3 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
3		eqid 2730	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	vciOLD 30079	. 2 ⊢ (𝑊 ∈ CVec_OLD → (𝐺 ∈ AbelOp ∧ (2^nd ‘𝑊):(ℂ × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺((1(2^nd ‘𝑊)𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝐺(𝑦(2^nd ‘𝑊)(𝑥𝐺𝑧)) = ((𝑦(2^nd ‘𝑊)𝑥)𝐺(𝑦(2^nd ‘𝑊)𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)(2^nd ‘𝑊)𝑥) = ((𝑦(2^nd ‘𝑊)𝑥)𝐺(𝑧(2^nd ‘𝑊)𝑥)) ∧ ((𝑦 · 𝑧)(2^nd ‘𝑊)𝑥) = (𝑦(2^nd ‘𝑊)(𝑧(2^nd ‘𝑊)𝑥)))))))
5	4	simp1d 1140	1 ⊢ (𝑊 ∈ CVec_OLD → 𝐺 ∈ AbelOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 × cxp 5675 ran crn 5678 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 1^st c1st 7977 2^nd c2nd 7978 ℂcc 11112 1c1 11115 + caddc 11117 · cmul 11119 AbelOpcablo 30062 CVec_OLDcvc 30076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7416 df-1st 7979 df-2nd 7980 df-vc 30077

This theorem is referenced by: vcgrp 30088 nvablo 30134 ip0i 30343 ipdirilem 30347

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