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Theorem nvvc 25803
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 2400 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2400 . . 3  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
41, 2, 3nvvop 25797 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( +v `  U ) ,  ( .sOLD `  U ) >. )
5 eqid 2400 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
6 eqid 2400 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
7 eqid 2400 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
85, 2, 3, 6, 7nvi 25802 . . 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 1007 . 2  |-  ( U  e.  NrmCVec  ->  <. ( +v `  U ) ,  ( .sOLD `  U
) >.  e.  CVecOLD )
104, 9eqeltrd 2488 1  |-  ( U  e.  NrmCVec  ->  W  e.  CVecOLD )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   <.cop 3975   class class class wbr 4392   -->wf 5519   ` cfv 5523  (class class class)co 6232   1stc1st 6734   CCcc 9438   RRcr 9439   0cc0 9440    + caddc 9443    x. cmul 9445    <_ cle 9577   abscabs 13121   CVecOLDcvc 25733   NrmCVeccnv 25772   +vcpv 25773   BaseSetcba 25774   .sOLDcns 25775   0veccn0v 25776   normCVcnmcv 25778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-1st 6736  df-2nd 6737  df-vc 25734  df-nv 25780  df-va 25783  df-ba 25784  df-sm 25785  df-0v 25786  df-nmcv 25788
This theorem is referenced by:  nvablo  25804  nvsf  25807  nvscl  25816  nvsid  25817  nvsass  25818  nvdi  25820  nvdir  25821  nv2  25822  nv0  25827  nvsz  25828  nvinv  25829  phop  26028  ip0i  26035  ipdirilem  26039  hlvc  26104
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