Theorem nvtri 30599

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.)

Hypotheses

Ref	Expression
nvtri.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvtri.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvtri.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
nvtri	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))

Proof of Theorem nvtri

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nvtri.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
2		nvtri.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		eqid 2729	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	3	smfval 30534	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))
5	4	eqcomi 2738	. . . . . . 7 ⊢ (2nd ‘(1st ‘𝑈)) = ( ·𝑠OLD ‘𝑈)
6		eqid 2729	. . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		nvtri.6	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
8	1, 2, 5, 6, 7	nvi 30543	. . . . . 6 ⊢ (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‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
9	8	simp3d 1144	. . . . 5 ⊢ (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‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
10		simp3 1138	. . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st ‘𝑈))𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	10	ralimi 3066	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st ‘𝑈))𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
12	9, 11	syl 17	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
13		fvoveq1 7410	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
15	14	oveq1d 7402	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝑦)))
16	13, 15	breq12d 5120	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦))))
17		oveq2 7395	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
18	17	fveq2d 6862	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19		fveq2 6858	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
20	19	oveq2d 7403	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝐴) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝐵)))
21	18, 20	breq12d 5120	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
22	16, 21	rspc2v 3599	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
23	12, 22	syl5 34	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
24	23	3impia 1117	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))
25	24	3comr 1125	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595   class class class wbr 5107  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  ℂcc 11066  ℝcr 11067  0cc0 11068   + caddc 11071   · cmul 11073   ≤ cle 11209  abscabs 15200  CVec_OLDcvc 30487  NrmCVeccnv 30513   +_𝑣 cpv 30514  BaseSetcba 30515   ·_𝑠OLD cns 30516  0_veccn0v 30517  norm_CVcnmcv 30519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-1st 7968  df-2nd 7969  df-vc 30488  df-nv 30521  df-va 30524  df-ba 30525  df-sm 30526  df-0v 30527  df-nmcv 30529

This theorem is referenced by:  nvmtri  30600  nvabs  30601  nvge0  30602  imsmetlem  30619  vacn  30623