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.

Step	Hyp	Ref	Expression
1		nvvc.1	. . 3 |- W = ( 1st ` U )
2		eqid 2400	. . 3 |- ( +v ` U ) = ( +v ` U )
3		eqid 2400	. . 3 |- ( .sOLD ` U ) = ( .sOLD ` U )
4	1, 2, 3	nvvop 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 )
8	5, 2, 3, 6, 7	nvi 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 ) ) ) ) )
9	8	simp1d 1007	. 2 |- ( U e. NrmCVec -> <. ( +v ` U ) , ( .sOLD ` U ) >. e. CVecOLD )
10	4, 9	eqeltrd 2488	1 |- ( U e. NrmCVec -> W e. CVecOLD )

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   norm_CVcnmcv 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