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Theorem psmettri2 22334
Description: Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmettri2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Proof of Theorem psmettri2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6364 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2 ispsmet 22329 . . . . . . . 8 (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
31, 2syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
43ibi 256 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
54simprd 483 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
65r19.21bi 3081 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
76simprd 483 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
87ralrimiva 3115 . 2 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
9 oveq1 6803 . . . . 5 (𝑎 = 𝐴 → (𝑎𝐷𝑏) = (𝐴𝐷𝑏))
10 oveq2 6804 . . . . . 6 (𝑎 = 𝐴 → (𝑐𝐷𝑎) = (𝑐𝐷𝐴))
1110oveq1d 6811 . . . . 5 (𝑎 = 𝐴 → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)))
129, 11breq12d 4800 . . . 4 (𝑎 = 𝐴 → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏))))
13 oveq2 6804 . . . . 5 (𝑏 = 𝐵 → (𝐴𝐷𝑏) = (𝐴𝐷𝐵))
14 oveq2 6804 . . . . . 6 (𝑏 = 𝐵 → (𝑐𝐷𝑏) = (𝑐𝐷𝐵))
1514oveq2d 6812 . . . . 5 (𝑏 = 𝐵 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)))
1613, 15breq12d 4800 . . . 4 (𝑏 = 𝐵 → ((𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵))))
17 oveq1 6803 . . . . . 6 (𝑐 = 𝐶 → (𝑐𝐷𝐴) = (𝐶𝐷𝐴))
18 oveq1 6803 . . . . . 6 (𝑐 = 𝐶 → (𝑐𝐷𝐵) = (𝐶𝐷𝐵))
1917, 18oveq12d 6814 . . . . 5 (𝑐 = 𝐶 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) = ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
2019breq2d 4799 . . . 4 (𝑐 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
2112, 16, 20rspc3v 3475 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑎𝑋𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
22213comr 1119 . 2 ((𝐶𝑋𝐴𝑋𝐵𝑋) → (∀𝑎𝑋𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
238, 22mpan9 496 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351   class class class wbr 4787   × cxp 5248  wf 6026  cfv 6030  (class class class)co 6796  0cc0 10142  *cxr 10279  cle 10281   +𝑒 cxad 12149  PsMetcpsmet 19945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-xr 10284  df-psmet 19953
This theorem is referenced by:  psmetsym  22335  psmettri  22336  psmetge0  22337  psmetres2  22339  xblss2ps  22426  metideq  30276  metider  30277  pstmxmet  30280
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