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.

Step	Hyp	Ref	Expression
1		nvvc.1	. . 3 |- W = ( 1st ` U )
2		eqid 2454	. . 3 |- ( +v ` U ) = ( +v ` U )
3		eqid 2454	. . 3 |- ( .sOLD ` U ) = ( .sOLD ` U )
4	1, 2, 3	nvvop 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 )
8	5, 2, 3, 6, 7	nvi 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 ) ) ) ) )
9	8	simp1d 1000	. 2 |- ( U e. NrmCVec -> <. ( +v ` U ) , ( .sOLD ` U ) >. e. CVecOLD )
10	4, 9	eqeltrd 2542	1 |- ( U e. NrmCVec -> W e. CVecOLD )

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   norm_CVcnmcv 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