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Theorem nvzcl 21208
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 2296 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 nvzcl.6 . . 3  |-  Z  =  ( 0vec `  U
)
31, 20vfval 21178 . 2  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
41nvgrp 21189 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
5 nvzcl.1 . . . . 5  |-  X  =  ( BaseSet `  U )
65, 1bafval 21176 . . . 4  |-  X  =  ran  ( +v `  U )
7 eqid 2296 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
86, 7grpoidcl 20900 . . 3  |-  ( ( +v `  U )  e.  GrpOp  ->  (GId `  ( +v `  U ) )  e.  X )
94, 8syl 15 . 2  |-  ( U  e.  NrmCVec  ->  (GId `  ( +v `  U ) )  e.  X )
103, 9eqeltrd 2370 1  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271   GrpOpcgr 20869  GIdcgi 20870   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   0veccn0v 21160
This theorem is referenced by:  nvzs  21219  nvmeq0  21238  nvz0  21250  elimnv  21268  nvnd  21273  imsmetlem  21275  nvlmle  21281  dip0r  21309  dip0l  21310  sspz  21327  lno0  21350  lnomul  21354  nvo00  21355  nmosetn0  21359  nmooge0  21361  0oo  21383  0lno  21384  nmoo0  21385  blocni  21399  ubthlem1  21465  minvecolem1  21469  hl0cl  21497  hhshsslem2  21861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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