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Theorem abvtri 19593
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 19584 . . . . . 6 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐵 = (Base‘𝑅)
4 abvtri.p . . . . . . 7 + = (+g𝑅)
5 eqid 2819 . . . . . . 7 (.r𝑅) = (.r𝑅)
6 eqid 2819 . . . . . . 7 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 19582 . . . . . 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 269 . . . 4 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
10 simpr 487 . . . . . . 7 (((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1110ralimi 3158 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1211adantl 484 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
1312ralimi 3158 . . . 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 506 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
15 fvoveq1 7171 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
16 fveq2 6663 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1716oveq1d 7163 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑦)))
1815, 17breq12d 5070 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦))))
19 oveq2 7156 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
2019fveq2d 6667 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
21 fveq2 6663 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221oveq2d 7164 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) + (𝐹𝑦)) = ((𝐹𝑋) + (𝐹𝑌)))
2320, 22breq12d 5070 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹𝑋) + (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌))))
2418, 23rspc2v 3631 . . 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 1111 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7148  0cc0 10529   + caddc 10532   · cmul 10534  +∞cpnf 10664  cle 10668  [,)cico 12732  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  0gc0g 16705  Ringcrg 19289  AbsValcabv 19579
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-abv 19580
This theorem is referenced by:  abvsubtri  19598  abvres  19602  abvcxp  26183  qabvle  26193  ostth2lem2  26202  ostth3  26206
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