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Theorem nvtri 28456
 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 2824 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
43smfval 28391 . . . . . . . 8 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
54eqcomi 2833 . . . . . . 7 (2nd ‘(1st𝑈)) = ( ·𝑠OLD𝑈)
6 eqid 2824 . . . . . . 7 (0vec𝑈) = (0vec𝑈)
7 nvtri.6 . . . . . . 7 𝑁 = (normCV𝑈)
81, 2, 5, 6, 7nvi 28400 . . . . . 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 3155 . . . . 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 7172 . . . . . 6 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
14 fveq2 6661 . . . . . . 7 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
1514oveq1d 7164 . . . . . 6 (𝑥 = 𝐴 → ((𝑁𝑥) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝑦)))
1613, 15breq12d 5065 . . . . 5 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ↔ (𝑁‘(𝐴𝐺𝑦)) ≤ ((𝑁𝐴) + (𝑁𝑦))))
17 oveq2 7157 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1817fveq2d 6665 . . . . . 6 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
19 fveq2 6661 . . . . . . 7 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
2019oveq2d 7165 . . . . . 6 (𝑦 = 𝐵 → ((𝑁𝐴) + (𝑁𝑦)) = ((𝑁𝐴) + (𝑁𝐵)))
2118, 20breq12d 5065 . . . . 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ⟨cop 4556   class class class wbr 5052  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  ℂcc 10533  ℝcr 10534  0cc0 10535   + caddc 10538   · cmul 10540   ≤ cle 10674  abscabs 14593  CVecOLDcvc 28344  NrmCVeccnv 28370   +𝑣 cpv 28371  BaseSetcba 28372   ·𝑠OLD cns 28373  0veccn0v 28374  normCVcnmcv 28376 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-1st 7684  df-2nd 7685  df-vc 28345  df-nv 28378  df-va 28381  df-ba 28382  df-sm 28383  df-0v 28384  df-nmcv 28386 This theorem is referenced by:  nvmtri  28457  nvabs  28458  nvge0  28459  imsmetlem  28476  vacn  28480
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