Theorem nvtri 30702

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 2740	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	3	smfval 30637	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))
5	4	eqcomi 2749	. . . . . . 7 ⊢ (2nd ‘(1st ‘𝑈)) = ( ·𝑠OLD ‘𝑈)
6		eqid 2740	. . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		nvtri.6	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
8	1, 2, 5, 6, 7	nvi 30646	. . . . . 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 3089	. . . . 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 7471	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
15	14	oveq1d 7463	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝑦)))
16	13, 15	breq12d 5179	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦))))
17		oveq2 7456	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
18	17	fveq2d 6924	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19		fveq2 6920	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
20	19	oveq2d 7464	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝐴) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝐵)))
21	18, 20	breq12d 5179	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
22	16, 21	rspc2v 3646	. . . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⟨cop 4654   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  ℂcc 11182  ℝcr 11183  0cc0 11184   + caddc 11187   · cmul 11189   ≤ cle 11325  abscabs 15283  CVec_OLDcvc 30590  NrmCVeccnv 30616   +_𝑣 cpv 30617  BaseSetcba 30618   ·_𝑠OLD cns 30619  0_veccn0v 30620  norm_CVcnmcv 30622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-1st 8030  df-2nd 8031  df-vc 30591  df-nv 30624  df-va 30627  df-ba 30628  df-sm 30629  df-0v 30630  df-nmcv 30632

This theorem is referenced by:  nvmtri  30703  nvabs  30704  nvge0  30705  imsmetlem  30722  vacn  30726