Theorem nvtri 30759

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 2737	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	3	smfval 30694	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))
5	4	eqcomi 2746	. . . . . . 7 ⊢ (2nd ‘(1st ‘𝑈)) = ( ·𝑠OLD ‘𝑈)
6		eqid 2737	. . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		nvtri.6	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
8	1, 2, 5, 6, 7	nvi 30703	. . . . . 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 1145	. . . . 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 1139	. . . . . 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 3075	. . . . 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 7384	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14		fveq2 6835	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
15	14	oveq1d 7376	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝑦)))
16	13, 15	breq12d 5099	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦))))
17		oveq2 7369	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
18	17	fveq2d 6839	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19		fveq2 6835	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
20	19	oveq2d 7377	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝐴) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝐵)))
21	18, 20	breq12d 5099	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
22	16, 21	rspc2v 3576	. . . 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 1118	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))
25	24	3comr 1126	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574   class class class wbr 5086  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  ℂcc 11030  ℝcr 11031  0cc0 11032   + caddc 11035   · cmul 11037   ≤ cle 11174  abscabs 15190  CVec_OLDcvc 30647  NrmCVeccnv 30673   +_𝑣 cpv 30674  BaseSetcba 30675   ·_𝑠OLD cns 30676  0_veccn0v 30677  norm_CVcnmcv 30679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-1st 7936  df-2nd 7937  df-vc 30648  df-nv 30681  df-va 30684  df-ba 30685  df-sm 30686  df-0v 30687  df-nmcv 30689

This theorem is referenced by:  nvmtri  30760  nvabs  30761  nvge0  30762  imsmetlem  30779  vacn  30783