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Theorem nvgcl 29560
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 = ( +𝑣𝑈)
Assertion
Ref Expression
nvgcl ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem nvgcl
StepHypRef Expression
1 nvgcl.2 . . 3 𝐺 = ( +𝑣𝑈)
21nvgrp 29557 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3 nvgcl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
43, 1bafval 29544 . . 3 𝑋 = ran 𝐺
54grpocl 29440 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
62, 5syl3an1 1163 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7356  GrpOpcgr 29429  NrmCVeccnv 29524   +𝑣 cpv 29525  BaseSetcba 29526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-1st 7920  df-2nd 7921  df-grpo 29433  df-ablo 29485  df-vc 29499  df-nv 29532  df-va 29535  df-ba 29536  df-sm 29537  df-0v 29538  df-nmcv 29540
This theorem is referenced by:  nvmf  29585  nvpncan2  29593  nvaddsub4  29597  nvdif  29606  nvpi  29607  nvabs  29612  imsmetlem  29630  vacn  29634  ipval2lem2  29644  4ipval2  29648  lnocoi  29697  0lno  29730  blocnilem  29744  ip0i  29765  ip1ilem  29766  ip2i  29768  ipdirilem  29769  ipasslem10  29779  dipdi  29783  ip2dii  29784  pythi  29790  ipblnfi  29795  ubthlem2  29811  minvecolem2  29815  hhshsslem2  30208
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