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Theorem nvzcl 22115
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 2436 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 nvzcl.6 . . 3  |-  Z  =  ( 0vec `  U
)
31, 20vfval 22085 . 2  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
41nvgrp 22096 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
5 nvzcl.1 . . . . 5  |-  X  =  ( BaseSet `  U )
65, 1bafval 22083 . . . 4  |-  X  =  ran  ( +v `  U )
7 eqid 2436 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
86, 7grpoidcl 21805 . . 3  |-  ( ( +v `  U )  e.  GrpOp  ->  (GId `  ( +v `  U ) )  e.  X )
94, 8syl 16 . 2  |-  ( U  e.  NrmCVec  ->  (GId `  ( +v `  U ) )  e.  X )
103, 9eqeltrd 2510 1  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454   GrpOpcgr 21774  GIdcgi 21775   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   0veccn0v 22067
This theorem is referenced by:  nvzs  22126  nvmeq0  22145  nvz0  22157  elimnv  22175  nvnd  22180  imsmetlem  22182  nvlmle  22188  dip0r  22216  dip0l  22217  sspz  22234  lno0  22257  lnomul  22261  nvo00  22262  nmosetn0  22266  nmooge0  22268  0oo  22290  0lno  22291  nmoo0  22292  blocni  22306  ubthlem1  22372  minvecolem1  22376  hl0cl  22404  hhshsslem2  22768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079
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