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Theorem nv0rid 28416
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 = ( +𝑣𝑈)
nv0id.6 𝑍 = (0vec𝑈)
Assertion
Ref Expression
nv0rid ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem nv0rid
StepHypRef Expression
1 nv0id.2 . . . . 5 𝐺 = ( +𝑣𝑈)
2 nv0id.6 . . . . 5 𝑍 = (0vec𝑈)
31, 20vfval 28387 . . . 4 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
43oveq2d 7156 . . 3 (𝑈 ∈ NrmCVec → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
54adantr 484 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
61nvgrp 28398 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
7 nv0id.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
87, 1bafval 28385 . . . 4 𝑋 = ran 𝐺
9 eqid 2822 . . . 4 (GId‘𝐺) = (GId‘𝐺)
108, 9grporid 28298 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
116, 10sylan 583 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
125, 11eqtrd 2857 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  GrpOpcgr 28270  GIdcgi 28271  NrmCVeccnv 28365   +𝑣 cpv 28366  BaseSetcba 28367  0veccn0v 28369
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-1st 7675  df-2nd 7676  df-grpo 28274  df-gid 28275  df-ablo 28326  df-vc 28340  df-nv 28373  df-va 28376  df-ba 28377  df-sm 28378  df-0v 28379  df-nmcv 28381
This theorem is referenced by:  nvabs  28453  nvnd  28469  imsmetlem  28471  lnomul  28541  0lno  28571  ipdirilem  28610  hladdid  28684
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