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Theorem nvvc 24165
Description: The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvvc.1  |-  W  =  ( 1st `  U
)
Assertion
Ref Expression
nvvc  |-  ( U  e.  NrmCVec  ->  W  e.  CVecOLD )

Proof of Theorem nvvc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvvc.1 . . 3  |-  W  =  ( 1st `  U
)
2 eqid 2454 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2454 . . 3  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
41, 2, 3nvvop 24159 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( +v `  U ) ,  ( .sOLD `  U ) >. )
5 eqid 2454 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
6 eqid 2454 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
7 eqid 2454 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
85, 2, 3, 6, 7nvi 24164 . . 3  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .sOLD `  U ) >.  e.  CVecOLD 
/\  ( normCV `  U
) : ( BaseSet `  U ) --> RR  /\  A. x  e.  ( BaseSet `  U ) ( ( ( ( normCV `  U
) `  x )  =  0  ->  x  =  ( 0vec `  U
) )  /\  A. y  e.  CC  (
( normCV `  U ) `  ( y ( .sOLD `  U ) x ) )  =  ( ( abs `  y
)  x.  ( (
normCV
`  U ) `  x ) )  /\  A. y  e.  ( BaseSet `  U ) ( (
normCV
`  U ) `  ( x ( +v
`  U ) y ) )  <_  (
( ( normCV `  U
) `  x )  +  ( ( normCV `  U ) `  y
) ) ) ) )
98simp1d 1000 . 2  |-  ( U  e.  NrmCVec  ->  <. ( +v `  U ) ,  ( .sOLD `  U
) >.  e.  CVecOLD )
104, 9eqeltrd 2542 1  |-  ( U  e.  NrmCVec  ->  W  e.  CVecOLD )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   <.cop 3994   class class class wbr 4403   -->wf 5525   ` cfv 5529  (class class class)co 6203   1stc1st 6688   CCcc 9394   RRcr 9395   0cc0 9396    + caddc 9399    x. cmul 9401    <_ cle 9533   abscabs 12844   CVecOLDcvc 24095   NrmCVeccnv 24134   +vcpv 24135   BaseSetcba 24136   .sOLDcns 24137   0veccn0v 24138   normCVcnmcv 24140
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-1st 6690  df-2nd 6691  df-vc 24096  df-nv 24142  df-va 24145  df-ba 24146  df-sm 24147  df-0v 24148  df-nmcv 24150
This theorem is referenced by:  nvablo  24166  nvsf  24169  nvscl  24178  nvsid  24179  nvsass  24180  nvdi  24182  nvdir  24183  nv2  24184  nv0  24189  nvsz  24190  nvinv  24191  phop  24390  ip0i  24397  ipdirilem  24401  hlvc  24466
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