Theorem nvgcl 28883

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 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem nvgcl

Step	Hyp	Ref	Expression
1		nvgcl.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2	1	nvgrp 28880	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvgcl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28867	. . 3 ⊢ 𝑋 = ran 𝐺
5	4	grpocl 28763	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
6	2, 5	syl3an1 1161	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  GrpOpcgr 28752  NrmCVeccnv 28847   +_𝑣 cpv 28848  BaseSetcba 28849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-1st 7804  df-2nd 7805  df-grpo 28756  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-nmcv 28863

This theorem is referenced by:  nvmf  28908  nvpncan2  28916  nvaddsub4  28920  nvdif  28929  nvpi  28930  nvabs  28935  imsmetlem  28953  vacn  28957  ipval2lem2  28967  4ipval2  28971  lnocoi  29020  0lno  29053  blocnilem  29067  ip0i  29088  ip1ilem  29089  ip2i  29091  ipdirilem  29092  ipasslem10  29102  dipdi  29106  ip2dii  29107  pythi  29113  ipblnfi  29118  ubthlem2  29134  minvecolem2  29138  hhshsslem2  29531