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Theorem nv0lid 28398
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 28368 . . . 4 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
43oveq1d 7148 . . 3 (𝑈 ∈ NrmCVec → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
54adantr 483 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
61nvgrp 28379 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
7 nv0id.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
87, 1bafval 28366 . . . 4 𝑋 = ran 𝐺
9 eqid 2820 . . . 4 (GId‘𝐺) = (GId‘𝐺)
108, 9grpolid 28278 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
116, 10sylan 582 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
125, 11eqtrd 2855 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cfv 6331  (class class class)co 7133  GrpOpcgr 28251  GIdcgi 28252  NrmCVeccnv 28346   +𝑣 cpv 28347  BaseSetcba 28348  0veccn0v 28350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-1st 7667  df-2nd 7668  df-grpo 28255  df-gid 28256  df-ablo 28307  df-vc 28321  df-nv 28354  df-va 28357  df-ba 28358  df-sm 28359  df-0v 28360  df-nmcv 28362
This theorem is referenced by:  nvpncan2  28415  nvmeq0  28420  imsmetlem  28452  ipdirilem  28591
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