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Theorem nvvc 21187
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.  CVec OLD )

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 2296 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2296 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
41, 2, 3nvvop 21181 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( +v `  U ) ,  ( .s OLD `  U ) >. )
5 eqid 2296 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
6 eqid 2296 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
7 eqid 2296 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
85, 2, 3, 6, 7nvi 21186 . . 3  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD  /\  ( normCV `  U ) : ( BaseSet `  U ) --> RR  /\  A. x  e.  ( BaseSet `  U )
( ( ( (
normCV
`  U ) `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( ( normCV `  U ) `  (
y ( .s OLD `  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 967 . 2  |-  ( U  e.  NrmCVec  ->  <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD )
104, 9eqeltrd 2370 1  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    <_ cle 8884   abscabs 11735   CVec
OLDcvc 21117   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   0veccn0v 21160   normCVcnmcv 21162
This theorem is referenced by:  nvablo  21188  nvsf  21191  nvscl  21200  nvsid  21201  nvsass  21202  nvdi  21204  nvdir  21205  nv2  21206  nv0  21211  nvsz  21212  nvinv  21213  phop  21412  ip0i  21419  ipdirilem  21423  hlvc  21488
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-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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