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Theorem nvzcl 25727
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 2454 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 nvzcl.6 . . 3  |-  Z  =  ( 0vec `  U
)
31, 20vfval 25697 . 2  |-  ( U  e.  NrmCVec  ->  Z  =  (GId
`  ( +v `  U ) ) )
41nvgrp 25708 . . 3  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
5 nvzcl.1 . . . . 5  |-  X  =  ( BaseSet `  U )
65, 1bafval 25695 . . . 4  |-  X  =  ran  ( +v `  U )
7 eqid 2454 . . . 4  |-  (GId `  ( +v `  U ) )  =  (GId `  ( +v `  U ) )
86, 7grpoidcl 25417 . . 3  |-  ( ( +v `  U )  e.  GrpOp  ->  (GId `  ( +v `  U ) )  e.  X )
94, 8syl 16 . 2  |-  ( U  e.  NrmCVec  ->  (GId `  ( +v `  U ) )  e.  X )
103, 9eqeltrd 2542 1  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570   GrpOpcgr 25386  GIdcgi 25387   NrmCVeccnv 25675   +vcpv 25676   BaseSetcba 25677   0veccn0v 25679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-nmcv 25691
This theorem is referenced by:  nvzs  25738  nvmeq0  25757  nvz0  25769  elimnv  25787  nvnd  25792  imsmetlem  25794  nvlmle  25800  dip0r  25828  dip0l  25829  sspz  25846  lno0  25869  lnomul  25873  nvo00  25874  nmosetn0  25878  nmooge0  25880  0oo  25902  0lno  25903  nmoo0  25904  blocni  25918  ubthlem1  25984  minvecolem1  25988  hl0cl  26016  hhshsslem2  26382
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