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Theorem nvgcl 21489
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 21486 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvgcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 21473 . . 3  |-  X  =  ran  G
54grpocl 21178 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
62, 5syl3an1 1216 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 935    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   GrpOpcgr 21164   NrmCVeccnv 21453   +vcpv 21454   BaseSetcba 21455
This theorem is referenced by:  nvmf  21517  nvsubadd  21526  nvpncan2  21527  nvaddsub4  21532  nvdif  21544  nvpi  21545  nvabs  21552  imsmetlem  21572  nvelbl2  21576  vacn  21580  ipval2lem2  21590  4ipval2  21594  sspival  21627  lnocoi  21648  0lno  21681  blocnilem  21695  ip0i  21716  ip1ilem  21717  ip2i  21719  ipdirilem  21720  ipasslem10  21730  dipdi  21734  ip2dii  21735  pythi  21741  sspph  21746  ipblnfi  21747  ubthlem2  21763  minvecolem2  21767  hhshsslem2  22158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-1st 6249  df-2nd 6250  df-grpo 21169  df-ablo 21260  df-vc 21415  df-nv 21461  df-va 21464  df-ba 21465  df-sm 21466  df-0v 21467  df-nmcv 21469
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