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Theorem nvgcl 25286
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 25283 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvgcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 25270 . . 3  |-  X  =  ran  G
54grpocl 24975 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
62, 5syl3an1 1261 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   GrpOpcgr 24961   NrmCVeccnv 25250   +vcpv 25251   BaseSetcba 25252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-1st 6785  df-2nd 6786  df-grpo 24966  df-ablo 25057  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-0v 25264  df-nmcv 25266
This theorem is referenced by:  nvmf  25314  nvsubadd  25323  nvpncan2  25324  nvaddsub4  25329  nvdif  25341  nvpi  25342  nvabs  25349  imsmetlem  25369  nvelbl2  25373  vacn  25377  ipval2lem2  25387  4ipval2  25391  sspival  25424  lnocoi  25445  0lno  25478  blocnilem  25492  ip0i  25513  ip1ilem  25514  ip2i  25516  ipdirilem  25517  ipasslem10  25527  dipdi  25531  ip2dii  25532  pythi  25538  sspph  25543  ipblnfi  25544  ubthlem2  25560  minvecolem2  25564  hhshsslem2  25957
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