Theorem nvtri 29612

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 2736	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4	3	smfval 29547	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))
5	4	eqcomi 2745	. . . . . . 7 ⊢ (2nd ‘(1st ‘𝑈)) = ( ·𝑠OLD ‘𝑈)
6		eqid 2736	. . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
7		nvtri.6	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
8	1, 2, 5, 6, 7	nvi 29556	. . . . . 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 3086	. . . . 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 7380	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
15	14	oveq1d 7372	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝑦)))
16	13, 15	breq12d 5118	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦))))
17		oveq2 7365	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
18	17	fveq2d 6846	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19		fveq2 6842	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
20	19	oveq2d 7373	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝐴) + (𝑁‘𝑦)) = ((𝑁‘𝐴) + (𝑁‘𝐵)))
21	18, 20	breq12d 5118	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁‘𝐴) + (𝑁‘𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))))
22	16, 21	rspc2v 3590	. . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ⟨cop 4592   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920  ℂcc 11049  ℝcr 11050  0cc0 11051   + caddc 11054   · cmul 11056   ≤ cle 11190  abscabs 15119  CVec_OLDcvc 29500  NrmCVeccnv 29526   +_𝑣 cpv 29527  BaseSetcba 29528   ·_𝑠OLD cns 29529  0_veccn0v 29530  norm_CVcnmcv 29532

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-1st 7921  df-2nd 7922  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-nmcv 29542

This theorem is referenced by:  nvmtri  29613  nvabs  29614  nvge0  29615  imsmetlem  29632  vacn  29636