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Theorem abvtri 20840
Description: An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a 𝐴 = (AbsVal‘𝑅)
abvf.b 𝐵 = (Base‘𝑅)
abvtri.p + = (+g𝑅)
Assertion
Ref Expression
abvtri ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))

Proof of Theorem abvtri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . . 7 𝐴 = (AbsVal‘𝑅)
21abvrcl 20831 . . . . . 6 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐵 = (Base‘𝑅)
4 abvtri.p . . . . . . 7 + = (+g𝑅)
5 eqid 2735 . . . . . . 7 (.r𝑅) = (.r𝑅)
6 eqid 2735 . . . . . . 7 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 20829 . . . . . 6 (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
82, 7syl 17 . . . . 5 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 267 . . . 4 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
10 simpr 484 . . . . . . 7 (((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1110ralimi 3081 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1211adantl 481 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1312ralimi 3081 . . . 4 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
149, 13simpl2im 503 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
15 fvoveq1 7454 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
16 fveq2 6907 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1716oveq1d 7446 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑦)))
1815, 17breq12d 5161 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦))))
19 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
2019fveq2d 6911 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
21 fveq2 6907 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑌)))
2320, 22breq12d 5161 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2418, 23rspc2v 3633 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2514, 24syl5com 31 . 2 (𝐹𝐴 → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
26253impib 1115 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153   + caddc 11156   · cmul 11158  +∞cpnf 11290  cle 11294  [,)cico 13386  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486  Ringcrg 20251  AbsValcabv 20826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-abv 20827
This theorem is referenced by:  abvsubtri  20845  abvres  20849  abvcxp  27674  qabvle  27684  ostth2lem2  27693  ostth3  27697
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