Theorem nvvc 30616

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 2733	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2733	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	1, 2, 3	nvvop 30610	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩)
5		eqid 2733	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2733	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		eqid 2733	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
8	5, 2, 3, 6, 7	nvi 30615	. . 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 2833	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ⟨cop 4583   class class class wbr 5095  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  ℂcc 11015  ℝcr 11016  0cc0 11017   + caddc 11020   · cmul 11022   ≤ cle 11158  abscabs 15148  CVec_OLDcvc 30559  NrmCVeccnv 30585   +_𝑣 cpv 30586  BaseSetcba 30587   ·_𝑠OLD cns 30588  0_veccn0v 30589  norm_CVcnmcv 30591

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-1st 7930  df-2nd 7931  df-vc 30560  df-nv 30593  df-va 30596  df-ba 30597  df-sm 30598  df-0v 30599  df-nmcv 30601

This theorem is referenced by:  nvablo  30617  nvsf  30620  nvscl  30627  nvsid  30628  nvsass  30629  nvdi  30631  nvdir  30632  nv2  30633  nv0  30638  nvsz  30639  nvinv  30640  phop  30819  ip0i  30826  ipdirilem  30830  hlvc  30894