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Theorem nv0rid 29575
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 29546 . . . 4 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
43oveq2d 7372 . . 3 (𝑈 ∈ NrmCVec → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
54adantr 481 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = (𝐴𝐺(GId‘𝐺)))
61nvgrp 29557 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
7 nv0id.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
87, 1bafval 29544 . . . 4 𝑋 = ran 𝐺
9 eqid 2736 . . . 4 (GId‘𝐺) = (GId‘𝐺)
108, 9grporid 29457 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
116, 10sylan 580 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
125, 11eqtrd 2776 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7356  GrpOpcgr 29429  GIdcgi 29430  NrmCVeccnv 29524   +𝑣 cpv 29525  BaseSetcba 29526  0veccn0v 29528
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 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
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 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-oprab 7360  df-1st 7920  df-2nd 7921  df-grpo 29433  df-gid 29434  df-ablo 29485  df-vc 29499  df-nv 29532  df-va 29535  df-ba 29536  df-sm 29537  df-0v 29538  df-nmcv 29540
This theorem is referenced by:  nvabs  29612  nvnd  29628  imsmetlem  29630  lnomul  29700  0lno  29730  ipdirilem  29769  hladdid  29843
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