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Theorem vcablo 31036

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 2762	. . 3 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
3		eqid 2762	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	vciOLD 31028	. 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 1160	1 ⊢ (𝑊 ∈ CVec_OLD → 𝐺 ∈ AbelOp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 × cxp 5657 ran crn 5660 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 1^st c1st 7987 2^nd c2nd 7988 ℂcc 11125 1c1 11128 + caddc 11130 · cmul 11132 AbelOpcablo 31011 CVec_OLDcvc 31025

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7419 df-1st 7989 df-2nd 7990 df-vc 31026

This theorem is used by: vcgrp 31037 nvablo 31083 ip0i 31292 ipdirilem 31296

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