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Theorem nvzcl 25205
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 2467 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 nvzcl.6 . . 3  |-  Z  =  ( 0vec `  U
)
31, 20vfval 25175 . 2  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
41nvgrp 25186 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
5 nvzcl.1 . . . . 5  |-  X  =  ( BaseSet `  U )
65, 1bafval 25173 . . . 4  |-  X  =  ran  ( +v `  U )
7 eqid 2467 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
86, 7grpoidcl 24895 . . 3  |-  ( ( +v `  U )  e.  GrpOp  ->  (GId `  ( +v `  U ) )  e.  X )
94, 8syl 16 . 2  |-  ( U  e.  NrmCVec  ->  (GId `  ( +v `  U ) )  e.  X )
103, 9eqeltrd 2555 1  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586   GrpOpcgr 24864  GIdcgi 24865   NrmCVeccnv 25153   +vcpv 25154   BaseSetcba 25155   0veccn0v 25157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gid 24870  df-ablo 24960  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-nmcv 25169
This theorem is referenced by:  nvzs  25216  nvmeq0  25235  nvz0  25247  elimnv  25265  nvnd  25270  imsmetlem  25272  nvlmle  25278  dip0r  25306  dip0l  25307  sspz  25324  lno0  25347  lnomul  25351  nvo00  25352  nmosetn0  25356  nmooge0  25358  0oo  25380  0lno  25381  nmoo0  25382  blocni  25396  ubthlem1  25462  minvecolem1  25466  hl0cl  25494  hhshsslem2  25860
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