Theorem nvvc 30967

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 2763	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2763	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	1, 2, 3	nvvop 30961	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩)
5		eqid 2763	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2763	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		eqid 2763	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
8	5, 2, 3, 6, 7	nvi 30966	. . 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 1160	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩ ∈ CVecOLD)
10	4, 9	eqeltrd 2863	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟨cop 4595   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  ℂcc 11093  ℝcr 11094  0cc0 11095   + caddc 11098   · cmul 11100   ≤ cle 11239  abscabs 15281  CVec_OLDcvc 30910  NrmCVeccnv 30936   +_𝑣 cpv 30937  BaseSetcba 30938   ·_𝑠OLD cns 30939  0_veccn0v 30940  norm_CVcnmcv 30942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-1st 7982  df-2nd 7983  df-vc 30911  df-nv 30944  df-va 30947  df-ba 30948  df-sm 30949  df-0v 30950  df-nmcv 30952

This theorem is referenced by:  nvablo  30968  nvsf  30971  nvscl  30978  nvsid  30979  nvsass  30980  nvdi  30982  nvdir  30983  nv2  30984  nv0  30989  nvsz  30990  nvinv  30991  phop  31170  ip0i  31177  ipdirilem  31181  hlvc  31245