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Theorem nvtri 28097
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 2777 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
43smfval 28032 . . . . . . . 8 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
54eqcomi 2786 . . . . . . 7 (2nd ‘(1st𝑈)) = ( ·𝑠OLD𝑈)
6 eqid 2777 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
7 nvtri.6 . . . . . . 7 𝑁 = (normCV𝑈)
81, 2, 5, 6, 7nvi 28041 . . . . . 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 1135 . . . . 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 1129 . . . . . 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 3133 . . . . 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 fvoveq1 6945 . . . . . 6 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14 fveq2 6446 . . . . . . 7 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
1514oveq1d 6937 . . . . . 6 (𝑥 = 𝐴 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝑦)))
1613, 15breq12d 4899 . . . . 5 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦))))
17 oveq2 6930 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1817fveq2d 6450 . . . . . 6 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19 fveq2 6446 . . . . . . 7 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
2019oveq2d 6938 . . . . . 6 (𝑦 = 𝐵 → ((𝑁𝐴) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝐵)))
2118, 20breq12d 4899 . . . . 5 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2216, 21rspc2v 3523 . . . 4 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2312, 22syl5 34 . . 3 ((𝐴𝑋𝐵𝑋) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
24233impia 1106 . 2 ((𝐴𝑋𝐵𝑋𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
25243comr 1116 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wral 3089  cop 4403   class class class wbr 4886  wf 6131  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  cc 10270  cr 10271  0cc0 10272   + caddc 10275   · cmul 10277  cle 10412  abscabs 14381  CVecOLDcvc 27985  NrmCVeccnv 28011   +𝑣 cpv 28012  BaseSetcba 28013   ·𝑠OLD cns 28014  0veccn0v 28015  normCVcnmcv 28017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-1st 7445  df-2nd 7446  df-vc 27986  df-nv 28019  df-va 28022  df-ba 28023  df-sm 28024  df-0v 28025  df-nmcv 28027
This theorem is referenced by:  nvmtri  28098  nvabs  28099  nvge0  28100  imsmetlem  28117  vacn  28121
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