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Theorem nvtri 27834
 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 = ( +𝑣𝑈)
nvtri.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvtri ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))

Proof of Theorem nvtri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvtri.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
2 nvtri.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
3 eqid 2760 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
43smfval 27769 . . . . . . . 8 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
54eqcomi 2769 . . . . . . 7 (2nd ‘(1st𝑈)) = ( ·𝑠OLD𝑈)
6 eqid 2760 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
7 nvtri.6 . . . . . . 7 𝑁 = (normCV𝑈)
81, 2, 5, 6, 7nvi 27778 . . . . . 6 (𝑈 ∈ NrmCVec → (⟨𝐺, (2nd ‘(1st𝑈))⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st𝑈))𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
98simp3d 1139 . . . . 5 (𝑈 ∈ NrmCVec → ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st𝑈))𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))
10 simp3 1133 . . . . . 6 ((((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st𝑈))𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
1110ralimi 3090 . . . . 5 (∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = (0vec𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦(2nd ‘(1st𝑈))𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))) → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
129, 11syl 17 . . . 4 (𝑈 ∈ NrmCVec → ∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))
13 oveq1 6820 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
1413fveq2d 6356 . . . . . 6 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
15 fveq2 6352 . . . . . . 7 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
1615oveq1d 6828 . . . . . 6 (𝑥 = 𝐴 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝑦)))
1714, 16breq12d 4817 . . . . 5 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦))))
18 oveq2 6821 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1918fveq2d 6356 . . . . . 6 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
20 fveq2 6352 . . . . . . 7 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
2120oveq2d 6829 . . . . . 6 (𝑦 = 𝐵 → ((𝑁𝐴) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝐵)))
2219, 21breq12d 4817 . . . . 5 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2317, 22rspc2v 3461 . . . 4 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2412, 23syl5 34 . . 3 ((𝐴𝑋𝐵𝑋) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
25243impia 1110 . 2 ((𝐴𝑋𝐵𝑋𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
26253comr 1120 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ⟨cop 4327   class class class wbr 4804  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  ℂcc 10126  ℝcr 10127  0cc0 10128   + caddc 10131   · cmul 10133   ≤ cle 10267  abscabs 14173  CVecOLDcvc 27722  NrmCVeccnv 27748   +𝑣 cpv 27749  BaseSetcba 27750   ·𝑠OLD cns 27751  0veccn0v 27752  normCVcnmcv 27754 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-1st 7333  df-2nd 7334  df-vc 27723  df-nv 27756  df-va 27759  df-ba 27760  df-sm 27761  df-0v 27762  df-nmcv 27764 This theorem is referenced by:  nvmtri  27835  nvabs  27836  nvge0  27837  imsmetlem  27854  vacn  27858
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