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Theorem nvzcl 30653
Description: Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvzcl.1 𝑋 = (BaseSet‘𝑈)
nvzcl.6 𝑍 = (0vec𝑈)
Assertion
Ref Expression
nvzcl (𝑈 ∈ NrmCVec → 𝑍𝑋)

Proof of Theorem nvzcl
StepHypRef Expression
1 eqid 2737 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
2 nvzcl.6 . . 3 𝑍 = (0vec𝑈)
31, 20vfval 30625 . 2 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣𝑈)))
41nvgrp 30636 . . 3 (𝑈 ∈ NrmCVec → ( +𝑣𝑈) ∈ GrpOp)
5 nvzcl.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
65, 1bafval 30623 . . . 4 𝑋 = ran ( +𝑣𝑈)
7 eqid 2737 . . . 4 (GId‘( +𝑣𝑈)) = (GId‘( +𝑣𝑈))
86, 7grpoidcl 30533 . . 3 (( +𝑣𝑈) ∈ GrpOp → (GId‘( +𝑣𝑈)) ∈ 𝑋)
94, 8syl 17 . 2 (𝑈 ∈ NrmCVec → (GId‘( +𝑣𝑈)) ∈ 𝑋)
103, 9eqeltrd 2841 1 (𝑈 ∈ NrmCVec → 𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  GrpOpcgr 30508  GIdcgi 30509  NrmCVeccnv 30603   +𝑣 cpv 30604  BaseSetcba 30605  0veccn0v 30607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ablo 30564  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619
This theorem is referenced by:  nvmeq0  30677  nvz0  30687  elimnv  30702  nvnd  30707  imsmetlem  30709  dip0r  30736  dip0l  30737  sspz  30754  lno0  30775  lnomul  30779  nvo00  30780  nmosetn0  30784  nmooge0  30786  0oo  30808  0lno  30809  nmoo0  30810  blocni  30824  ubthlem1  30889  minvecolem1  30893  hl0cl  30921  hhshsslem2  31287
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