Theorem nvzcl 28044

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 2825	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		nvzcl.6	. . 3 ⊢ 𝑍 = (0vec‘𝑈)
3	1, 2	0vfval 28016	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈)))
4	1	nvgrp 28027	. . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp)
5		nvzcl.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	5, 1	bafval 28014	. . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
7		eqid 2825	. . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈))
8	6, 7	grpoidcl 27924	. . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋)
9	4, 8	syl 17	. 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋)
10	3, 9	eqeltrd 2906	1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ‘cfv 6123  GrpOpcgr 27899  GIdcgi 27900  NrmCVeccnv 27994   +_𝑣 cpv 27995  BaseSetcba 27996  0_veccn0v 27998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-1st 7428  df-2nd 7429  df-grpo 27903  df-gid 27904  df-ablo 27955  df-vc 27969  df-nv 28002  df-va 28005  df-ba 28006  df-sm 28007  df-0v 28008  df-nmcv 28010

This theorem is referenced by:  nvmeq0  28068  nvz0  28078  elimnv  28093  nvnd  28098  imsmetlem  28100  dip0r  28127  dip0l  28128  sspz  28145  lno0  28166  lnomul  28170  nvo00  28171  nmosetn0  28175  nmooge0  28177  0oo  28199  0lno  28200  nmoo0  28201  blocni  28215  ubthlem1  28281  minvecolem1  28285  hl0cl  28313  hhshsslem2  28680