Theorem nvvc 28014

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 2825	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2825	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	1, 2, 3	nvvop 28008	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩)
5		eqid 2825	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2825	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		eqid 2825	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
8	5, 2, 3, 6, 7	nvi 28013	. . 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 1176	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩ ∈ CVecOLD)
10	4, 9	eqeltrd 2906	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ⟨cop 4403   class class class wbr 4873  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  1^st c1st 7426  ℂcc 10250  ℝcr 10251  0cc0 10252   + caddc 10255   · cmul 10257   ≤ cle 10392  abscabs 14351  CVec_OLDcvc 27957  NrmCVeccnv 27983   +_𝑣 cpv 27984  BaseSetcba 27985   ·_𝑠OLD cns 27986  0_veccn0v 27987  norm_CVcnmcv 27989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-1st 7428  df-2nd 7429  df-vc 27958  df-nv 27991  df-va 27994  df-ba 27995  df-sm 27996  df-0v 27997  df-nmcv 27999

This theorem is referenced by:  nvablo  28015  nvsf  28018  nvscl  28025  nvsid  28026  nvsass  28027  nvdi  28029  nvdir  28030  nv2  28031  nv0  28036  nvsz  28037  nvinv  28038  phop  28217  ip0i  28224  ipdirilem  28228  hlvc  28293