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Theorem nvgcl 22052
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  =  ( +v `  U
)
Assertion
Ref Expression
nvgcl  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem nvgcl
StepHypRef Expression
1 nvgcl.2 . . 3  |-  G  =  ( +v `  U
)
21nvgrp 22049 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvgcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 22036 . . 3  |-  X  =  ran  G
54grpocl 21741 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
62, 5syl3an1 1217 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018
This theorem is referenced by:  nvmf  22080  nvsubadd  22089  nvpncan2  22090  nvaddsub4  22095  nvdif  22107  nvpi  22108  nvabs  22115  imsmetlem  22135  nvelbl2  22139  vacn  22143  ipval2lem2  22153  4ipval2  22157  sspival  22190  lnocoi  22211  0lno  22244  blocnilem  22258  ip0i  22279  ip1ilem  22280  ip2i  22282  ipdirilem  22283  ipasslem10  22293  dipdi  22297  ip2dii  22298  pythi  22304  sspph  22309  ipblnfi  22310  ubthlem2  22326  minvecolem2  22330  hhshsslem2  22721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-1st 6308  df-2nd 6309  df-grpo 21732  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032
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