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Theorem nvgcl 28401
 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 28398 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3 nvgcl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
43, 1bafval 28385 . . 3 𝑋 = ran 𝐺
54grpocl 28281 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
62, 5syl3an1 1160 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ‘cfv 6344  (class class class)co 7146  GrpOpcgr 28270  NrmCVeccnv 28365   +𝑣 cpv 28366  BaseSetcba 28367 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-1st 7681  df-2nd 7682  df-grpo 28274  df-ablo 28326  df-vc 28340  df-nv 28373  df-va 28376  df-ba 28377  df-sm 28378  df-0v 28379  df-nmcv 28381 This theorem is referenced by:  nvmf  28426  nvpncan2  28434  nvaddsub4  28438  nvdif  28447  nvpi  28448  nvabs  28453  imsmetlem  28471  vacn  28475  ipval2lem2  28485  4ipval2  28489  lnocoi  28538  0lno  28571  blocnilem  28585  ip0i  28606  ip1ilem  28607  ip2i  28609  ipdirilem  28610  ipasslem10  28620  dipdi  28624  ip2dii  28625  pythi  28631  ipblnfi  28636  ubthlem2  28652  minvecolem2  28656  hhshsslem2  29049
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