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Theorem nvgcl 27992
Description: Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvgcl.1 𝑋 = (BaseSet‘𝑈)
nvgcl.2 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
nvgcl ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem nvgcl
StepHypRef Expression
1 nvgcl.2 . . 3 𝐺 = ( +𝑣𝑈)
21nvgrp 27989 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3 nvgcl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
43, 1bafval 27976 . . 3 𝑋 = ran 𝐺
54grpocl 27872 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
62, 5syl3an1 1203 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  cfv 6099  (class class class)co 6876  GrpOpcgr 27861  NrmCVeccnv 27956   +𝑣 cpv 27957  BaseSetcba 27958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-1st 7399  df-2nd 7400  df-grpo 27865  df-ablo 27917  df-vc 27931  df-nv 27964  df-va 27967  df-ba 27968  df-sm 27969  df-0v 27970  df-nmcv 27972
This theorem is referenced by:  nvmf  28017  nvpncan2  28025  nvaddsub4  28029  nvdif  28038  nvpi  28039  nvabs  28044  imsmetlem  28062  vacn  28066  ipval2lem2  28076  4ipval2  28080  lnocoi  28129  0lno  28162  blocnilem  28176  ip0i  28197  ip1ilem  28198  ip2i  28200  ipdirilem  28201  ipasslem10  28211  dipdi  28215  ip2dii  28216  pythi  28222  sspphOLD  28227  ipblnfi  28228  ubthlem2  28244  minvecolem2  28248  hhshsslem2  28642
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