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Theorem nvtri 30603
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 2726 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
43smfval 30538 . . . . . . . 8 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
54eqcomi 2735 . . . . . . 7 (2nd ‘(1st𝑈)) = ( ·𝑠OLD𝑈)
6 eqid 2726 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
7 nvtri.6 . . . . . . 7 𝑁 = (normCV𝑈)
81, 2, 5, 6, 7nvi 30547 . . . . . 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 1141 . . . . 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 1135 . . . . . 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 3073 . . . . 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 7447 . . . . . 6 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14 fveq2 6901 . . . . . . 7 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
1514oveq1d 7439 . . . . . 6 (𝑥 = 𝐴 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝑦)))
1613, 15breq12d 5166 . . . . 5 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦))))
17 oveq2 7432 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1817fveq2d 6905 . . . . . 6 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19 fveq2 6901 . . . . . . 7 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
2019oveq2d 7440 . . . . . 6 (𝑦 = 𝐵 → ((𝑁𝐴) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝐵)))
2118, 20breq12d 5166 . . . . 5 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2216, 21rspc2v 3619 . . . 4 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
2312, 22syl5 34 . . 3 ((𝐴𝑋𝐵𝑋) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵))))
24233impia 1114 . 2 ((𝐴𝑋𝐵𝑋𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
25243comr 1122 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cop 4639   class class class wbr 5153  wf 6550  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  cc 11156  cr 11157  0cc0 11158   + caddc 11161   · cmul 11163  cle 11299  abscabs 15239  CVecOLDcvc 30491  NrmCVeccnv 30517   +𝑣 cpv 30518  BaseSetcba 30519   ·𝑠OLD cns 30520  0veccn0v 30521  normCVcnmcv 30523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-1st 8003  df-2nd 8004  df-vc 30492  df-nv 30525  df-va 30528  df-ba 30529  df-sm 30530  df-0v 30531  df-nmcv 30533
This theorem is referenced by:  nvmtri  30604  nvabs  30605  nvge0  30606  imsmetlem  30623  vacn  30627
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