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Theorem isxmet 23677
Description: Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
isxmet (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐷   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem isxmet
Dummy variables 𝑑 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3463 . . . . 5 (𝑋𝐴𝑋 ∈ V)
2 xpeq12 5658 . . . . . . . . 9 ((𝑡 = 𝑋𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
32anidms 567 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
43oveq2d 7373 . . . . . . 7 (𝑡 = 𝑋 → (ℝ*m (𝑡 × 𝑡)) = (ℝ*m (𝑋 × 𝑋)))
5 raleq 3309 . . . . . . . . . 10 (𝑡 = 𝑋 → (∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65anbi2d 629 . . . . . . . . 9 (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
76raleqbi1dv 3307 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
87raleqbi1dv 3307 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3424 . . . . . 6 (𝑡 = 𝑋 → {𝑑 ∈ (ℝ*m (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
10 df-xmet 20789 . . . . . 6 ∞Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 7390 . . . . . . 7 (ℝ*m (𝑋 × 𝑋)) ∈ V
1211rabex 5289 . . . . . 6 {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6948 . . . . 5 (𝑋 ∈ V → (∞Met‘𝑋) = {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
141, 13syl 17 . . . 4 (𝑋𝐴 → (∞Met‘𝑋) = {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
1514eleq2d 2823 . . 3 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 7363 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1716eqeq1d 2738 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
1817bibi1d 343 . . . . . 6 (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
19 oveq 7363 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 7363 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 7375 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2216, 21breq12d 5118 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2322ralbidv 3174 . . . . . 6 (𝑑 = 𝐷 → (∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2418, 23anbi12d 631 . . . . 5 (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
25242ralbidv 3212 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3645 . . 3 (𝐷 ∈ {𝑑 ∈ (ℝ*m (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2715, 26bitrdi 286 . 2 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28 xrex 12912 . . . 4 * ∈ V
29 sqxpexg 7689 . . . 4 (𝑋𝐴 → (𝑋 × 𝑋) ∈ V)
30 elmapg 8778 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 587 . . 3 (𝑋𝐴 → (𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3231anbi1d 630 . 2 (𝑋𝐴 → ((𝐷 ∈ (ℝ*m (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
3327, 32bitrd 278 1 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445   class class class wbr 5105   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  0cc0 11051  *cxr 11188  cle 11190   +𝑒 cxad 13031  ∞Metcxmet 20781
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-xr 11193  df-xmet 20789
This theorem is referenced by:  isxmetd  23679  xmetf  23682  ismet2  23686  xmeteq0  23691  xmettri2  23693  imasf1oxmet  23728  pstmxmet  32478
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