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Theorem ispsmet 22156
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 3243 . . . . 5 (𝑋𝑉𝑋 ∈ V)
2 id 22 . . . . . . . . 9 (𝑢 = 𝑋𝑢 = 𝑋)
32sqxpeqd 5175 . . . . . . . 8 (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
43oveq2d 6706 . . . . . . 7 (𝑢 = 𝑋 → (ℝ*𝑚 (𝑢 × 𝑢)) = (ℝ*𝑚 (𝑋 × 𝑋)))
5 raleq 3168 . . . . . . . . . 10 (𝑢 = 𝑋 → (∀𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65raleqbi1dv 3176 . . . . . . . . 9 (𝑢 = 𝑋 → (∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
76anbi2d 740 . . . . . . . 8 (𝑢 = 𝑋 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
87raleqbi1dv 3176 . . . . . . 7 (𝑢 = 𝑋 → (∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3226 . . . . . 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 19786 . . . . . 6 PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 6718 . . . . . . 7 (ℝ*𝑚 (𝑋 × 𝑋)) ∈ V
1211rabex 4845 . . . . . 6 {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6321 . . . . 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 2716 . . 3 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 6696 . . . . . . 7 (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
1716eqeq1d 2653 . . . . . 6 (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18 oveq 6696 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19 oveq 6696 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 6696 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 6708 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2218, 21breq12d 4698 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23222ralbidv 3018 . . . . . 6 (𝑑 = 𝐷 → (∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2417, 23anbi12d 747 . . . . 5 (𝑑 = 𝐷 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2524ralbidv 3015 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3396 . . 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 276 . 2 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28 xrex 11867 . . . 4 * ∈ V
29 sqxpexg 7005 . . . 4 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
30 elmapg 7912 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 696 . . 3 (𝑋𝑉 → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3231anbi1d 741 . 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 268 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  *cxr 10111  cle 10113   +𝑒 cxad 11982  PsMetcpsmet 19778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-xr 10116  df-psmet 19786
This theorem is referenced by:  psmetdmdm  22157  psmetf  22158  psmet0  22160  psmettri2  22161  psmetres2  22166  xmetpsmet  22200
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