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Theorem nv0lid 29423
Description: The zero vector is a left 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
nv0lid ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)

Proof of Theorem nv0lid
StepHypRef Expression
1 nv0id.2 . . . . 5 𝐺 = ( +𝑣𝑈)
2 nv0id.6 . . . . 5 𝑍 = (0vec𝑈)
31, 20vfval 29393 . . . 4 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
43oveq1d 7366 . . 3 (𝑈 ∈ NrmCVec → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
54adantr 481 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
61nvgrp 29404 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
7 nv0id.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
87, 1bafval 29391 . . . 4 𝑋 = ran 𝐺
9 eqid 2736 . . . 4 (GId‘𝐺) = (GId‘𝐺)
108, 9grpolid 29303 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
116, 10sylan 580 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
125, 11eqtrd 2776 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  GrpOpcgr 29276  GIdcgi 29277  NrmCVeccnv 29371   +𝑣 cpv 29372  BaseSetcba 29373  0veccn0v 29375
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-1st 7913  df-2nd 7914  df-grpo 29280  df-gid 29281  df-ablo 29332  df-vc 29346  df-nv 29379  df-va 29382  df-ba 29383  df-sm 29384  df-0v 29385  df-nmcv 29387
This theorem is referenced by:  nvpncan2  29440  nvmeq0  29445  imsmetlem  29477  ipdirilem  29616
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