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Theorem nvgcl 26125
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 26122 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvgcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 26109 . . 3  |-  X  =  ran  G
54grpocl 25814 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
62, 5syl3an1 1297 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 982    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296   GrpOpcgr 25800   NrmCVeccnv 26089   +vcpv 26090   BaseSetcba 26091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-1st 6798  df-2nd 6799  df-grpo 25805  df-ablo 25896  df-vc 26051  df-nv 26097  df-va 26100  df-ba 26101  df-sm 26102  df-0v 26103  df-nmcv 26105
This theorem is referenced by:  nvmf  26153  nvsubadd  26162  nvpncan2  26163  nvaddsub4  26168  nvdif  26180  nvpi  26181  nvabs  26188  imsmetlem  26208  nvelbl2  26212  vacn  26216  ipval2lem2  26226  4ipval2  26230  sspival  26263  lnocoi  26284  0lno  26317  blocnilem  26331  ip0i  26352  ip1ilem  26353  ip2i  26355  ipdirilem  26356  ipasslem10  26366  dipdi  26370  ip2dii  26371  pythi  26377  sspph  26382  ipblnfi  26383  ubthlem2  26399  minvecolem2  26403  hhshsslem2  26795
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