Theorem nvvc 8234

Description: The vector space component of a normed complex vector space.

Hypothesis

Ref	Expression
nvvc.1	|- W = (1st` U)

Assertion

Ref	Expression
nvvc	|- (U e. NrmCVec -> W e. CVec)

Proof of Theorem nvvc

Step	Hyp	Ref	Expression
1		nvvc.1	. . 3 |- W = (1st` U)
2	1	fveq2i 3727	. . . 4 |- (1st` W) = (1st` (1st` U))
3		eqid 1475	. . . . 5 |- (+v` U) = (+v` U)
4	3	vafval 8222	. . . 4 |- (+v` U) = (1st` (1st` U))
5	2, 4	eqtr4 1498	. . 3 |- (1st` W) = (+v` U)
6	1	fveq2i 3727	. . . 4 |- (2nd` W) = (2nd` (1st` U))
7		eqid 1475	. . . . 5 |- (.s` U) = (.s` U)
8	7	smfval 8224	. . . 4 |- (.s` U) = (2nd` (1st` U))
9	6, 8	eqtr4 1498	. . 3 |- (2nd` W) = (.s` U)
10	1, 5, 9	nvvop 8228	. 2 |- (U e. NrmCVec -> W = <.(1st` W), (2nd` W)>.)
11		eqid 1475	. . . . . 6 |- (Base` U) = (Base` U)
12	11, 5	bafval 8223	. . . . 5 |- (Base` U) = ran (1st` W)
13	12	eqcomi 1479	. . . 4 |- ran (1st` W) = (Base` U)
14		eqid 1475	. . . . . 6 |- (0v` U) = (0v` U)
15	5, 14	0vfval 8225	. . . . 5 |- (0v` U) = (Id` (1st` W))
16	15	eqcomi 1479	. . . 4 |- (Id` (1st` W)) = (0v` U)
17		eqid 1475	. . . . . 6 |- (norm` U) = (norm` U)
18	17	nmfval 8226	. . . . 5 |- (norm` U) = (2nd` U)
19	18	eqcomi 1479	. . . 4 |- (2nd` U) = (norm` U)
20	13, 5, 9, 16, 19	nvi 8233	. . 3 |- (U e. NrmCVec -> (<.(1st` W), (2nd` W)>. e. CVec /\ (2nd` U):ran (1st` W)-->RR /\ A.x e. ran (1st` W)((((2nd` U)` x) = 0 -> x = (Id` (1st` W))) /\ A.y e. CC ((2nd` U)` (y(2nd` W)x)) = ((abs` y) x. ((2nd` U)` x)) /\ A.y e. ran (1st` W)((2nd` U)` (x(1st` W)y)) <_ (((2nd` U)` x) + ((2nd` U)` y)))))
21	20	3simp1d 794	. 2 |- (U e. NrmCVec -> <.(1st` W), (2nd` W)>. e. CVec)
22	10, 21	eqeltrd 1548	1 |- (U e. NrmCVec -> W e. CVec)

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  <.cop 2411   class class class wbr 2619  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  CCcc 5232  RRcr 5233  0cc0 5234   + caddc 5237   x. cmul 5239   <_ cle 5295  abscabs 6750  Idcgi 8034  CVeccvc 8164  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  0vcn0v 8207  normcnm 8209

This theorem is referenced by:  nvabl 8235  nvsf 8238  nvscl 8247  nvsid 8248  nvsass 8249  nvdi 8251  nvdir 8252  nv2 8253  nv0 8258  nvsz 8259  nvinv 8260  nvoprne 8306  sm1cnilem 8347  ipid 8363  phop 8477  ip0i 8484  ipdirilem 8488  hlvc 8597

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219

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