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Theorem nvgcl 29027
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 = (BaseSetβ€˜π‘ˆ)
nvgcl.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
Assertion
Ref Expression
nvgcl ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)

Proof of Theorem nvgcl
StepHypRef Expression
1 nvgcl.2 . . 3 𝐺 = ( +𝑣 β€˜π‘ˆ)
21nvgrp 29024 . 2 (π‘ˆ ∈ NrmCVec β†’ 𝐺 ∈ GrpOp)
3 nvgcl.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
43, 1bafval 29011 . . 3 𝑋 = ran 𝐺
54grpocl 28907 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
62, 5syl3an1 1163 1 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  β€˜cfv 6458  (class class class)co 7307  GrpOpcgr 28896  NrmCVeccnv 28991   +𝑣 cpv 28992  BaseSetcba 28993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-1st 7863  df-2nd 7864  df-grpo 28900  df-ablo 28952  df-vc 28966  df-nv 28999  df-va 29002  df-ba 29003  df-sm 29004  df-0v 29005  df-nmcv 29007
This theorem is referenced by:  nvmf  29052  nvpncan2  29060  nvaddsub4  29064  nvdif  29073  nvpi  29074  nvabs  29079  imsmetlem  29097  vacn  29101  ipval2lem2  29111  4ipval2  29115  lnocoi  29164  0lno  29197  blocnilem  29211  ip0i  29232  ip1ilem  29233  ip2i  29235  ipdirilem  29236  ipasslem10  29246  dipdi  29250  ip2dii  29251  pythi  29257  ipblnfi  29262  ubthlem2  29278  minvecolem2  29282  hhshsslem2  29675
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