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Theorem ispsmet 22889
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 3489 . . . . 5 (𝑋𝑉𝑋 ∈ V)
2 id 22 . . . . . . . . 9 (𝑢 = 𝑋𝑢 = 𝑋)
32sqxpeqd 5560 . . . . . . . 8 (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
43oveq2d 7146 . . . . . . 7 (𝑢 = 𝑋 → (ℝ*m (𝑢 × 𝑢)) = (ℝ*m (𝑋 × 𝑋)))
5 raleq 3390 . . . . . . . . . 10 (𝑢 = 𝑋 → (∀𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65raleqbi1dv 3388 . . . . . . . . 9 (𝑢 = 𝑋 → (∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
76anbi2d 631 . . . . . . . 8 (𝑢 = 𝑋 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
87raleqbi1dv 3388 . . . . . . 7 (𝑢 = 𝑋 → (∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3462 . . . . . 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 20512 . . . . . 6 PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑢 × 𝑢)) ∣ ∀𝑥𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑢𝑧𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 7163 . . . . . . 7 (ℝ*m (𝑋 × 𝑋)) ∈ V
1211rabex 5208 . . . . . 6 {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6741 . . . . 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 2897 . . 3 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 7136 . . . . . . 7 (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
1716eqeq1d 2823 . . . . . 6 (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18 oveq 7136 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19 oveq 7136 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 7136 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 7148 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2218, 21breq12d 5052 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23222ralbidv 3187 . . . . . 6 (𝑑 = 𝐷 → (∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2417, 23anbi12d 633 . . . . 5 (𝑑 = 𝐷 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2524ralbidv 3185 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3657 . . 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 290 . 2 (𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28 xrex 12364 . . . 4 * ∈ V
29 sqxpexg 7452 . . . 4 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
30 elmapg 8394 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 590 . . 3 (𝑋𝑉 → (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3231anbi1d 632 . 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 282 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 209  wa 399   = wceq 1538  wcel 2115  wral 3126  {crab 3130  Vcvv 3471   class class class wbr 5039   × cxp 5526  wf 6324  cfv 6328  (class class class)co 7130  m cmap 8381  0cc0 10514  *cxr 10651  cle 10653   +𝑒 cxad 12483  PsMetcpsmet 20504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-xr 10656  df-psmet 20512
This theorem is referenced by:  psmetdmdm  22890  psmetf  22891  psmet0  22893  psmettri2  22894  psmetres2  22899  xmetpsmet  22933
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