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Theorem abvtri 20100
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 20091 . . . . . 6 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐵 = (Base‘𝑅)
4 abvtri.p . . . . . . 7 + = (+g𝑅)
5 eqid 2738 . . . . . . 7 (.r𝑅) = (.r𝑅)
6 eqid 2738 . . . . . . 7 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 20089 . . . . . 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 266 . . . 4 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
10 simpr 485 . . . . . . 7 (((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1110ralimi 3087 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1211adantl 482 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1312ralimi 3087 . . . 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 504 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
15 fvoveq1 7290 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
16 fveq2 6766 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1716oveq1d 7282 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑦)))
1815, 17breq12d 5086 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦))))
19 oveq2 7275 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
2019fveq2d 6770 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
21 fveq2 6766 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221oveq2d 7283 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑌)))
2320, 22breq12d 5086 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2418, 23rspc2v 3569 . . 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064   class class class wbr 5073  wf 6422  cfv 6426  (class class class)co 7267  0cc0 10881   + caddc 10884   · cmul 10886  +∞cpnf 11016  cle 11020  [,)cico 13091  Basecbs 16922  +gcplusg 16972  .rcmulr 16973  0gc0g 17160  Ringcrg 19793  AbsValcabv 20086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-map 8604  df-abv 20087
This theorem is referenced by:  abvsubtri  20105  abvres  20109  abvcxp  26773  qabvle  26783  ostth2lem2  26792  ostth3  26796
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