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Theorem ispsmet 22103
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 3210 . . . . 5 (𝑋𝑉𝑋 ∈ V)
2 id 22 . . . . . . . . 9 (𝑢 = 𝑋𝑢 = 𝑋)
32sqxpeqd 5139 . . . . . . . 8 (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
43oveq2d 6663 . . . . . . 7 (𝑢 = 𝑋 → (ℝ*𝑚 (𝑢 × 𝑢)) = (ℝ*𝑚 (𝑋 × 𝑋)))
5 raleq 3136 . . . . . . . . . 10 (𝑢 = 𝑋 → (∀𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65raleqbi1dv 3144 . . . . . . . . 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 3144 . . . . . . 7 (𝑢 = 𝑋 → (∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3193 . . . . . 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 19732 . . . . . 6 PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 6675 . . . . . . 7 (ℝ*𝑚 (𝑋 × 𝑋)) ∈ V
1211rabex 4811 . . . . . 6 {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6280 . . . . 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 2686 . . 3 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 6653 . . . . . . 7 (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
1716eqeq1d 2623 . . . . . 6 (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18 oveq 6653 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19 oveq 6653 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 6653 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 6665 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2218, 21breq12d 4664 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23222ralbidv 2988 . . . . . 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 2985 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3361 . . 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 11826 . . . 4 * ∈ V
29 sqxpexg 6960 . . . 4 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
30 elmapg 7867 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 695 . . 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 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  Vcvv 3198   class class class wbr 4651   × cxp 5110  wf 5882  cfv 5886  (class class class)co 6647  𝑚 cmap 7854  0cc0 9933  *cxr 10070  cle 10072   +𝑒 cxad 11941  PsMetcpsmet 19724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-xr 10075  df-psmet 19732
This theorem is referenced by:  psmetdmdm  22104  psmetf  22105  psmet0  22107  psmettri2  22108  psmetres2  22113  xmetpsmet  22147
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