Theorem nvvc 8186

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 3718	. . . 4 |- (1st` W) = (1st` (1st` U))
3		eqid 1473	. . . . 5 |- (+v` U) = (+v` U)
4	3	vafval 8174	. . . 4 |- (+v` U) = (1st` (1st` U))
5	2, 4	eqtr4 1495	. . 3 |- (1st` W) = (+v` U)
6	1	fveq2i 3718	. . . 4 |- (2nd` W) = (2nd` (1st` U))
7		eqid 1473	. . . . 5 |- (.s` U) = (.s` U)
8	7	smfval 8176	. . . 4 |- (.s` U) = (2nd` (1st` U))
9	6, 8	eqtr4 1495	. . 3 |- (2nd` W) = (.s` U)
10	1, 5, 9	nvvop 8180	. 2 |- (U e. NrmCVec -> W = <.(1st` W), (2nd` W)>.)
11		eqid 1473	. . . . . 6 |- (Base` U) = (Base` U)
12	11, 5	bafval 8175	. . . . 5 |- (Base` U) = ran (1st` W)
13	12	eqcomi 1476	. . . 4 |- ran (1st` W) = (Base` U)
14		eqid 1473	. . . . . 6 |- (0v` U) = (0v` U)
15	5, 14	0vfval 8177	. . . . 5 |- (0v` U) = (Id` (1st` W))
16	15	eqcomi 1476	. . . 4 |- (Id` (1st` W)) = (0v` U)
17		eqid 1473	. . . . . 6 |- (norm` U) = (norm` U)
18	17	nmfval 8178	. . . . 5 |- (norm` U) = (2nd` U)
19	18	eqcomi 1476	. . . 4 |- (2nd` U) = (norm` U)
20	13, 5, 9, 16, 19	nvi 8185	. . 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 793	. 2 |- (U e. NrmCVec -> <.(1st` W), (2nd` W)>. e. CVec)
22	10, 21	eqeltrd 1545	1 |- (U e. NrmCVec -> W e. CVec)

Syntax hints:   -> wi 3   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  <.cop 2407   class class class wbr 2614  ran crn 3166  -->wf 3173  ` cfv 3177  (class class class)co 3954  1stc1st 4067  2ndc2nd 4068  CCcc 5212  RRcr 5213  0cc0 5214   + caddc 5217   x. cmul 5219   <_ cle 5275  abscabs 6689  Idcgi 7984  CVeccvc 8116  NrmCVeccnv 8155  +vcpv 8156  Basecba 8157  .scns 8158  0vcn0v 8159  normcnm 8161

This theorem is referenced by:  nvabl 8187  nvsf 8190  nvscl 8199  nvsid 8200  nvsass 8201  nvdi 8203  nvdir 8204  nv2 8205  nv0 8210  nvsz 8211  nvinv 8212  sm1cnilem 8294  ipid 8310  phop 8421  ip0i 8428  ipdirilem 8432  hlvc 8541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-grp 7987  df-gid 7988  df-nv 8163  df-va 8166  df-ba 8167  df-sm 8168  df-0v 8169  df-nm 8171

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