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Theorem ispsmet 21822
Description: Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
ispsmet (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑋   𝑥,𝐷,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem ispsmet
Dummy variables 𝑢 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . . . . 5 (𝑋𝑉𝑋 ∈ V)
2 id 22 . . . . . . . . 9 (𝑢 = 𝑋𝑢 = 𝑋)
32sqxpeqd 4959 . . . . . . . 8 (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
43oveq2d 6442 . . . . . . 7 (𝑢 = 𝑋 → (ℝ*𝑚 (𝑢 × 𝑢)) = (ℝ*𝑚 (𝑋 × 𝑋)))
5 raleq 3019 . . . . . . . . . 10 (𝑢 = 𝑋 → (∀𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65raleqbi1dv 3027 . . . . . . . . 9 (𝑢 = 𝑋 → (∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
76anbi2d 735 . . . . . . . 8 (𝑢 = 𝑋 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
87raleqbi1dv 3027 . . . . . . 7 (𝑢 = 𝑋 → (∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3072 . . . . . 6 (𝑢 = 𝑋 → {𝑑 ∈ (ℝ*𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
10 df-psmet 19463 . . . . . 6 PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 6454 . . . . . . 7 (ℝ*𝑚 (𝑋 × 𝑋)) ∈ V
1211rabex 4639 . . . . . 6 {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6075 . . . . 5 (𝑋 ∈ V → (PsMet‘𝑋) = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
141, 13syl 17 . . . 4 (𝑋𝑉 → (PsMet‘𝑋) = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
1514eleq2d 2577 . . 3 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 6432 . . . . . . 7 (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
1716eqeq1d 2516 . . . . . 6 (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18 oveq 6432 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19 oveq 6432 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 6432 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 6444 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2218, 21breq12d 4494 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23222ralbidv 2876 . . . . . 6 (𝑑 = 𝐷 → (∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2417, 23anbi12d 742 . . . . 5 (𝑑 = 𝐷 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2524ralbidv 2873 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3235 . . 3 (𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2715, 26syl6bb 274 . 2 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28 xrex 11571 . . . 4 * ∈ V
29 sqxpexg 6737 . . . 4 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
30 elmapg 7633 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 693 . . 3 (𝑋𝑉 → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3231anbi1d 736 . 2 (𝑋𝑉 → ((𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
3327, 32bitrd 266 1 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  {crab 2804  Vcvv 3077   class class class wbr 4481   × cxp 4930  wf 5685  cfv 5689  (class class class)co 6426  𝑚 cmap 7620  0cc0 9691  *cxr 9828  cle 9830   +𝑒 cxad 11686  PsMetcpsmet 19455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-map 7622  df-xr 9833  df-psmet 19463
This theorem is referenced by:  psmetdmdm  21823  psmetf  21824  psmet0  21826  psmettri2  21827  psmetres2  21832  xmetpsmet  21865
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