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Theorem nvzcl 24135
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  e.  NrmCVec  ->  Z  e.  X
)

Proof of Theorem nvzcl
StepHypRef Expression
1 eqid 2450 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 nvzcl.6 . . 3  |-  Z  =  ( 0vec `  U
)
31, 20vfval 24105 . 2  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
41nvgrp 24116 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
5 nvzcl.1 . . . . 5  |-  X  =  ( BaseSet `  U )
65, 1bafval 24103 . . . 4  |-  X  =  ran  ( +v `  U )
7 eqid 2450 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
86, 7grpoidcl 23825 . . 3  |-  ( ( +v `  U )  e.  GrpOp  ->  (GId `  ( +v `  U ) )  e.  X )
94, 8syl 16 . 2  |-  ( U  e.  NrmCVec  ->  (GId `  ( +v `  U ) )  e.  X )
103, 9eqeltrd 2536 1  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   ` cfv 5502   GrpOpcgr 23794  GIdcgi 23795   NrmCVeccnv 24083   +vcpv 24084   BaseSetcba 24085   0veccn0v 24087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-1st 6663  df-2nd 6664  df-grpo 23799  df-gid 23800  df-ablo 23890  df-vc 24045  df-nv 24091  df-va 24094  df-ba 24095  df-sm 24096  df-0v 24097  df-nmcv 24099
This theorem is referenced by:  nvzs  24146  nvmeq0  24165  nvz0  24177  elimnv  24195  nvnd  24200  imsmetlem  24202  nvlmle  24208  dip0r  24236  dip0l  24237  sspz  24254  lno0  24277  lnomul  24281  nvo00  24282  nmosetn0  24286  nmooge0  24288  0oo  24310  0lno  24311  nmoo0  24312  blocni  24326  ubthlem1  24392  minvecolem1  24396  hl0cl  24424  hhshsslem2  24790
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