Theorem nvtri

Description: Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006) (Revised by Mario Carneiro, 21-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvtri.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvtri.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	nvtri.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
Assertion	nvtri	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nvtri.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvtri.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvtri.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
4		eqid	⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 )
5	4	smfval	⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( 2nd ‘ ( 1st ‘ 𝑈 ) )
6	5	eqcomi	⊢ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) = ( ·𝑠OLD ‘ 𝑈 )
7		eqid	⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ 𝑈 )
8	1 2 6 7 3	nvi	⊢ ( 𝑈 ∈ NrmCVec → ( ⟨ 𝐺 , ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ⟩ ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( 2nd ‘ ( 1st ‘ 𝑈 ) ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )
9	8	simp3d	⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( 2nd ‘ ( 1st ‘ 𝑈 ) ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) )
10		simp3	⊢ ( ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( 2nd ‘ ( 1st ‘ 𝑈 ) ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
11	10	ralimi	⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( 2nd ‘ ( 1st ‘ 𝑈 ) ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
12	9 11	syl	⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
13		fvoveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) = ( 𝑁 ‘ ( 𝐴 𝐺 𝑦 ) ) )
14		fveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑁 ‘ 𝑥 ) = ( 𝑁 ‘ 𝐴 ) )
15	14	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) = ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝑦 ) ) )
16	13 15	breq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ↔ ( 𝑁 ‘ ( 𝐴 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝑦 ) ) ) )
17		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 𝑦 ) = ( 𝐴 𝐺 𝐵 ) )
18	17	fveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑁 ‘ ( 𝐴 𝐺 𝑦 ) ) = ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) )
19		fveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑁 ‘ 𝑦 ) = ( 𝑁 ‘ 𝐵 ) )
20	19	oveq2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝑦 ) ) = ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )
21	18 20	breq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑁 ‘ ( 𝐴 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝑦 ) ) ↔ ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) ) )
22	16 21	rspc2v	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 𝐺 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) ) )
23	12 22	syl5	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑈 ∈ NrmCVec → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) ) )
24	23	3impia	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑈 ∈ NrmCVec ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )
25	24	3comr	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )