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Theorem abvtri 19052
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 . . . . . . . 8 𝐴 = (AbsVal‘𝑅)
21abvrcl 19043 . . . . . . 7 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . . 8 𝐵 = (Base‘𝑅)
4 abvtri.p . . . . . . . 8 + = (+g𝑅)
5 eqid 2760 . . . . . . . 8 (.r𝑅) = (.r𝑅)
6 eqid 2760 . . . . . . . 8 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 19041 . . . . . . 7 (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 . . . . . 6 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 256 . . . . 5 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
109simprd 482 . . . 4 (𝐹𝐴 → ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
11 simpr 479 . . . . . . 7 (((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1211ralimi 3090 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1312adantl 473 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1413ralimi 3090 . . . 4 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1510, 14syl 17 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
16 oveq1 6821 . . . . . 6 (𝑥 = 𝑋 → (𝑥 + 𝑦) = (𝑋 + 𝑦))
1716fveq2d 6357 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
18 fveq2 6353 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1918oveq1d 6829 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑦)))
2017, 19breq12d 4817 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦))))
21 oveq2 6822 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
2221fveq2d 6357 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
23 fveq2 6353 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2423oveq2d 6830 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑌)))
2522, 24breq12d 4817 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2620, 25rspc2v 3461 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2715, 26syl5com 31 . 2 (𝐹𝐴 → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
28273impib 1109 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  wf 6045  cfv 6049  (class class class)co 6814  0cc0 10148   + caddc 10151   · cmul 10153  +∞cpnf 10283  cle 10287  [,)cico 12390  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  0gc0g 16322  Ringcrg 18767  AbsValcabv 19038
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-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
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-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  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-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-abv 19039
This theorem is referenced by:  abvsubtri  19057  abvres  19061  abvcxp  25524  qabvle  25534  ostth2lem2  25543  ostth3  25547
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