Theorem nv0rid 28418

Description: The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nv0id.1	⊢ 𝑋 = (BaseSet‘𝑈)
nv0id.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nv0id.6	⊢ 𝑍 = (0vec‘𝑈)

Assertion

Ref	Expression
nv0rid	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem nv0rid

Step	Hyp	Ref	Expression
1		nv0id.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2		nv0id.6	. . . . 5 ⊢ 𝑍 = (0vec‘𝑈)
3	1, 2	0vfval 28389	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
4	3	oveq2d 7151	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
5	4	adantr 484	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
6	1	nvgrp 28400	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
7		nv0id.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 1	bafval 28387	. . . 4 ⊢ 𝑋 = ran 𝐺
9		eqid 2798	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	8, 9	grporid 28300	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
11	6, 10	sylan 583	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
12	5, 11	eqtrd 2833	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  GrpOpcgr 28272  GIdcgi 28273  NrmCVeccnv 28367   +_𝑣 cpv 28368  BaseSetcba 28369  0_veccn0v 28371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-1st 7671  df-2nd 7672  df-grpo 28276  df-gid 28277  df-ablo 28328  df-vc 28342  df-nv 28375  df-va 28378  df-ba 28379  df-sm 28380  df-0v 28381  df-nmcv 28383

This theorem is referenced by:  nvabs  28455  nvnd  28471  imsmetlem  28473  lnomul  28543  0lno  28573  ipdirilem  28612  hladdid  28686