Theorem nvvc 30587

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 ∈ NrmCVec → 𝑊 ∈ 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 2731	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2731	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	1, 2, 3	nvvop 30581	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩)
5		eqid 2731	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2731	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		eqid 2731	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
8	5, 2, 3, 6, 7	nvi 30586	. . 3 ⊢ (𝑈 ∈ NrmCVec → (⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩ ∈ CVecOLD ∧ (normCV‘𝑈):(BaseSet‘𝑈)⟶ℝ ∧ ∀𝑥 ∈ (BaseSet‘𝑈)((((normCV‘𝑈)‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ ((normCV‘𝑈)‘(𝑦( ·𝑠OLD ‘𝑈)𝑥)) = ((abs‘𝑦) · ((normCV‘𝑈)‘𝑥)) ∧ ∀𝑦 ∈ (BaseSet‘𝑈)((normCV‘𝑈)‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ (((normCV‘𝑈)‘𝑥) + ((normCV‘𝑈)‘𝑦)))))
9	8	simp1d 1142	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩ ∈ CVecOLD)
10	4, 9	eqeltrd 2831	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⟨cop 4577   class class class wbr 5086  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  ℂcc 10999  ℝcr 11000  0cc0 11001   + caddc 11004   · cmul 11006   ≤ cle 11142  abscabs 15136  CVec_OLDcvc 30530  NrmCVeccnv 30556   +_𝑣 cpv 30557  BaseSetcba 30558   ·_𝑠OLD cns 30559  0_veccn0v 30560  norm_CVcnmcv 30562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-1st 7916  df-2nd 7917  df-vc 30531  df-nv 30564  df-va 30567  df-ba 30568  df-sm 30569  df-0v 30570  df-nmcv 30572

This theorem is referenced by:  nvablo  30588  nvsf  30591  nvscl  30598  nvsid  30599  nvsass  30600  nvdi  30602  nvdir  30603  nv2  30604  nv0  30609  nvsz  30610  nvinv  30611  phop  30790  ip0i  30797  ipdirilem  30801  hlvc  30865