Theorem nvi

Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006) (Revised by Mario Carneiro, 21-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	nvi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	nvi.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
	nvi.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
Assertion	nvi	⊢ ( 𝑈 ∈ NrmCVec → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nvi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		nvi.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
5		nvi.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
6		eqid	⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 )
7	6 5	nvop2	⊢ ( 𝑈 ∈ NrmCVec → 𝑈 = ⟨ ( 1st ‘ 𝑈 ) , 𝑁 ⟩ )
8	6 2 3	nvvop	⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) = ⟨ 𝐺 , 𝑆 ⟩ )
9	8	opeq1d	⊢ ( 𝑈 ∈ NrmCVec → ⟨ ( 1st ‘ 𝑈 ) , 𝑁 ⟩ = ⟨ ⟨ 𝐺 , 𝑆 ⟩ , 𝑁 ⟩ )
10	7 9	eqtrd	⊢ ( 𝑈 ∈ NrmCVec → 𝑈 = ⟨ ⟨ 𝐺 , 𝑆 ⟩ , 𝑁 ⟩ )
11		id	⊢ ( 𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec )
12	10 11	eqeltrrd	⊢ ( 𝑈 ∈ NrmCVec → ⟨ ⟨ 𝐺 , 𝑆 ⟩ , 𝑁 ⟩ ∈ NrmCVec )
13	1 2	bafval	⊢ 𝑋 = ran 𝐺
14		eqid	⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
15	13 14	isnv	⊢ ( ⟨ ⟨ 𝐺 , 𝑆 ⟩ , 𝑁 ⟩ ∈ NrmCVec ↔ ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
16	12 15	sylib	⊢ ( 𝑈 ∈ NrmCVec → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
17	2 4	0vfval	⊢ ( 𝑈 ∈ NrmCVec → 𝑍 = ( GId ‘ 𝐺 ) )
18	17	eqeq2d	⊢ ( 𝑈 ∈ NrmCVec → ( 𝑥 = 𝑍 ↔ 𝑥 = ( GId ‘ 𝐺 ) ) )
19	18	imbi2d	⊢ ( 𝑈 ∈ NrmCVec → ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ↔ ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ) )
20	19	3anbi1d	⊢ ( 𝑈 ∈ NrmCVec → ( ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ↔ ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
21	20	ralbidv	⊢ ( 𝑈 ∈ NrmCVec → ( ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
22	21	3anbi3d	⊢ ( 𝑈 ∈ NrmCVec → ( ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) ↔ ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( GId ‘ 𝐺 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) ) )
23	16 22	mpbird	⊢ ( 𝑈 ∈ NrmCVec → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )