Theorem nvzcl 28561

Description: Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvzcl.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvzcl.6	⊢ 𝑍 = (0vec‘𝑈)

Assertion

Ref	Expression
nvzcl	⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)

Proof of Theorem nvzcl

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		nvzcl.6	. . 3 ⊢ 𝑍 = (0vec‘𝑈)
3	1, 2	0vfval 28533	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈)))
4	1	nvgrp 28544	. . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp)
5		nvzcl.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	5, 1	bafval 28531	. . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
7		eqid 2738	. . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈))
8	6, 7	grpoidcl 28441	. . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋)
9	4, 8	syl 17	. 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋)
10	3, 9	eqeltrd 2833	1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  ‘cfv 6333  GrpOpcgr 28416  GIdcgi 28417  NrmCVeccnv 28511   +_𝑣 cpv 28512  BaseSetcba 28513  0_veccn0v 28515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-1st 7707  df-2nd 7708  df-grpo 28420  df-gid 28421  df-ablo 28472  df-vc 28486  df-nv 28519  df-va 28522  df-ba 28523  df-sm 28524  df-0v 28525  df-nmcv 28527

This theorem is referenced by:  nvmeq0  28585  nvz0  28595  elimnv  28610  nvnd  28615  imsmetlem  28617  dip0r  28644  dip0l  28645  sspz  28662  lno0  28683  lnomul  28687  nvo00  28688  nmosetn0  28692  nmooge0  28694  0oo  28716  0lno  28717  nmoo0  28718  blocni  28732  ubthlem1  28797  minvecolem1  28801  hl0cl  28829  hhshsslem2  29195