Theorem nvvc 31038

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 2765	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		eqid 2765	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	1, 2, 3	nvvop 31032	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨( +𝑣 ‘𝑈), ( ·𝑠OLD ‘𝑈)⟩)
5		eqid 2765	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2765	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		eqid 2765	. . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
8	5, 2, 3, 6, 7	nvi 31037	. . 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 2865	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ⟨cop 4597   class class class wbr 5111  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  ℂcc 11113  ℝcr 11114  0cc0 11115   + caddc 11118   · cmul 11120   ≤ cle 11259  abscabs 15309  CVec_OLDcvc 30981  NrmCVeccnv 31007   +_𝑣 cpv 31008  BaseSetcba 31009   ·_𝑠OLD cns 31010  0_veccn0v 31011  norm_CVcnmcv 31013

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-1st 7992  df-2nd 7993  df-vc 30982  df-nv 31015  df-va 31018  df-ba 31019  df-sm 31020  df-0v 31021  df-nmcv 31023

This theorem is used by:  nvablo  31039  nvsf  31042  nvscl  31049  nvsid  31050  nvsass  31051  nvdi  31053  nvdir  31054  nv2  31055  nv0  31060  nvsz  31061  nvinv  31062  phop  31241  ip0i  31248  ipdirilem  31252  hlvc  31316