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Theorem nvzcl 28344
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 2826 . . 3 ( +𝑣𝑈) = ( +𝑣𝑈)
2 nvzcl.6 . . 3 𝑍 = (0vec𝑈)
31, 20vfval 28316 . 2 (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣𝑈)))
41nvgrp 28327 . . 3 (𝑈 ∈ NrmCVec → ( +𝑣𝑈) ∈ GrpOp)
5 nvzcl.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
65, 1bafval 28314 . . . 4 𝑋 = ran ( +𝑣𝑈)
7 eqid 2826 . . . 4 (GId‘( +𝑣𝑈)) = (GId‘( +𝑣𝑈))
86, 7grpoidcl 28224 . . 3 (( +𝑣𝑈) ∈ GrpOp → (GId‘( +𝑣𝑈)) ∈ 𝑋)
94, 8syl 17 . 2 (𝑈 ∈ NrmCVec → (GId‘( +𝑣𝑈)) ∈ 𝑋)
103, 9eqeltrd 2918 1 (𝑈 ∈ NrmCVec → 𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cfv 6354  GrpOpcgr 28199  GIdcgi 28200  NrmCVeccnv 28294   +𝑣 cpv 28295  BaseSetcba 28296  0veccn0v 28298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-1st 7685  df-2nd 7686  df-grpo 28203  df-gid 28204  df-ablo 28255  df-vc 28269  df-nv 28302  df-va 28305  df-ba 28306  df-sm 28307  df-0v 28308  df-nmcv 28310
This theorem is referenced by:  nvmeq0  28368  nvz0  28378  elimnv  28393  nvnd  28398  imsmetlem  28400  dip0r  28427  dip0l  28428  sspz  28445  lno0  28466  lnomul  28470  nvo00  28471  nmosetn0  28475  nmooge0  28477  0oo  28499  0lno  28500  nmoo0  28501  blocni  28515  ubthlem1  28580  minvecolem1  28584  hl0cl  28612  hhshsslem2  28978
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