Theorem nvs

Description: Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvs.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvs.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	nvs.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
Assertion	nvs	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nvs.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvs.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		nvs.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
4		eqid	⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 )
5		eqid	⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ 𝑈 )
6	1 4 2 5 3	nvi	⊢ ( 𝑈 ∈ NrmCVec → ( ⟨ ( +𝑣 ‘ 𝑈 ) , 𝑆 ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
7	6	simp3d	⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) )
8		simp2	⊢ ( ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) )
9	8	ralimi	⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) )
10	7 9	syl	⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) )
11		oveq2	⊢ ( 𝑥 = 𝐵 → ( 𝑦 𝑆 𝑥 ) = ( 𝑦 𝑆 𝐵 ) )
12	11	fveq2d	⊢ ( 𝑥 = 𝐵 → ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( 𝑁 ‘ ( 𝑦 𝑆 𝐵 ) ) )
13		fveq2	⊢ ( 𝑥 = 𝐵 → ( 𝑁 ‘ 𝑥 ) = ( 𝑁 ‘ 𝐵 ) )
14	13	oveq2d	⊢ ( 𝑥 = 𝐵 → ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝐵 ) ) )
15	12 14	eqeq12d	⊢ ( 𝑥 = 𝐵 → ( ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ↔ ( 𝑁 ‘ ( 𝑦 𝑆 𝐵 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝐵 ) ) ) )
16		fvoveq1	⊢ ( 𝑦 = 𝐴 → ( 𝑁 ‘ ( 𝑦 𝑆 𝐵 ) ) = ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) )
17		fveq2	⊢ ( 𝑦 = 𝐴 → ( abs ‘ 𝑦 ) = ( abs ‘ 𝐴 ) )
18	17	oveq1d	⊢ ( 𝑦 = 𝐴 → ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) )
19	16 18	eqeq12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑁 ‘ ( 𝑦 𝑆 𝐵 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝐵 ) ) ↔ ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) ) )
20	15 19	rspc2v	⊢ ( ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 𝑆 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) → ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) ) )
21	10 20	syl5	⊢ ( ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ) → ( 𝑈 ∈ NrmCVec → ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) ) )
22	21	3impia	⊢ ( ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec ) → ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) )
23	22	3com13	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝑆 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( 𝑁 ‘ 𝐵 ) ) )