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Theorem abvtri 20332
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 20323 . . . . . 6 (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
4 abvtri.p . . . . . . 7 + = (+gβ€˜π‘…)
5 eqid 2733 . . . . . . 7 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6 eqid 2733 . . . . . . 7 (0gβ€˜π‘…) = (0gβ€˜π‘…)
71, 3, 4, 5, 6isabv 20321 . . . . . 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 486 . . . . . . 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 3083 . . . . . 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 483 . . . . 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 3083 . . . 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 505 . . 3 (𝐹 ∈ 𝐴 β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
15 fvoveq1 7384 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜(π‘₯ + 𝑦)) = (πΉβ€˜(𝑋 + 𝑦)))
16 fveq2 6846 . . . . . 6 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1716oveq1d 7376 . . . . 5 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦)))
1815, 17breq12d 5122 . . . 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 7369 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 + 𝑦) = (𝑋 + π‘Œ))
2019fveq2d 6850 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜(𝑋 + 𝑦)) = (πΉβ€˜(𝑋 + π‘Œ)))
21 fveq2 6846 . . . . . 6 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
2221oveq2d 7377 . . . . 5 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))
2320, 22breq12d 5122 . . . 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 3592 . . 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 1117 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  0cc0 11059   + caddc 11062   Β· cmul 11064  +∞cpnf 11194   ≀ cle 11198  [,)cico 13275  Basecbs 17091  +gcplusg 17141  .rcmulr 17142  0gc0g 17329  Ringcrg 19972  AbsValcabv 20318
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-abv 20319
This theorem is referenced by:  abvsubtri  20337  abvres  20341  abvcxp  26986  qabvle  26996  ostth2lem2  27005  ostth3  27009
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