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Theorem abvtri 20669
Description: An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a 𝐴 = (AbsValβ€˜π‘…)
abvf.b 𝐡 = (Baseβ€˜π‘…)
abvtri.p + = (+gβ€˜π‘…)
Assertion
Ref Expression
abvtri ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))

Proof of Theorem abvtri
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . . 7 𝐴 = (AbsValβ€˜π‘…)
21abvrcl 20660 . . . . . 6 (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
4 abvtri.p . . . . . . 7 + = (+gβ€˜π‘…)
5 eqid 2724 . . . . . . 7 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6 eqid 2724 . . . . . . 7 (0gβ€˜π‘…) = (0gβ€˜π‘…)
71, 3, 4, 5, 6isabv 20658 . . . . . 6 (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
82, 7syl 17 . . . . 5 (𝐹 ∈ 𝐴 β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
98ibi 267 . . . 4 (𝐹 ∈ 𝐴 β†’ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
10 simpr 484 . . . . . . 7 (((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
1110ralimi 3075 . . . . . 6 (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
1211adantl 481 . . . . 5 ((((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
1312ralimi 3075 . . . 4 (βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
149, 13simpl2im 503 . . 3 (𝐹 ∈ 𝐴 β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
15 fvoveq1 7425 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜(π‘₯ + 𝑦)) = (πΉβ€˜(𝑋 + 𝑦)))
16 fveq2 6882 . . . . . 6 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1716oveq1d 7417 . . . . 5 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦)))
1815, 17breq12d 5152 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝑋 + 𝑦)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦))))
19 oveq2 7410 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 + 𝑦) = (𝑋 + π‘Œ))
2019fveq2d 6886 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜(𝑋 + 𝑦)) = (πΉβ€˜(𝑋 + π‘Œ)))
21 fveq2 6882 . . . . . 6 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
2221oveq2d 7418 . . . . 5 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))
2320, 22breq12d 5152 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜(𝑋 + 𝑦)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ))))
2418, 23rspc2v 3615 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)) β†’ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ))))
2514, 24syl5com 31 . 2 (𝐹 ∈ 𝐴 β†’ ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ))))
26253impib 1113 1 ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   class class class wbr 5139  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  0cc0 11107   + caddc 11110   Β· cmul 11112  +∞cpnf 11244   ≀ cle 11248  [,)cico 13327  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  0gc0g 17390  Ringcrg 20134  AbsValcabv 20655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-abv 20656
This theorem is referenced by:  abvsubtri  20674  abvres  20678  abvcxp  27489  qabvle  27499  ostth2lem2  27508  ostth3  27512
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