Theorem ispsmet 23457

Description: Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
ispsmet	⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (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.

Step	Hyp	Ref	Expression
1		elex 3450	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
2		id 22	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → 𝑢 = 𝑋)
3	2	sqxpeqd 5621	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
4	3	oveq2d 7291	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (ℝ* ↑m (𝑢 × 𝑢)) = (ℝ* ↑m (𝑋 × 𝑋)))
5		raleq 3342	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
6	5	raleqbi1dv 3340	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
7	6	anbi2d 629	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
8	7	raleqbi1dv 3340	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
9	4, 8	rabeqbidv 3420	. . . . . 6 ⊢ (𝑢 = 𝑋 → {𝑑 ∈ (ℝ* ↑m (𝑢 × 𝑢)) ∣ ∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
10		df-psmet 20589	. . . . . 6 ⊢ PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑢 × 𝑢)) ∣ ∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11		ovex 7308	. . . . . . 7 ⊢ (ℝ* ↑m (𝑋 × 𝑋)) ∈ V
12	11	rabex 5256	. . . . . 6 ⊢ {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
13	9, 10, 12	fvmpt 6875	. . . . 5 ⊢ (𝑋 ∈ V → (PsMet‘𝑋) = {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
14	1, 13	syl 17	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (PsMet‘𝑋) = {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
15	14	eleq2d 2824	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16		oveq 7281	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
17	16	eqeq1d 2740	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18		oveq 7281	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19		oveq 7281	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20		oveq 7281	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
21	19, 20	oveq12d 7293	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
22	18, 21	breq12d 5087	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23	22	2ralbidv 3129	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
24	17, 23	anbi12d 631	. . . . 5 ⊢ (𝑑 = 𝐷 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
25	24	ralbidv 3112	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
26	25	elrab 3624	. . 3 ⊢ (𝐷 ∈ {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
27	15, 26	bitrdi 287	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28		xrex 12727	. . . 4 ⊢ ℝ* ∈ V
29		sqxpexg 7605	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
30		elmapg 8628	. . . 4 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
31	28, 29, 30	sylancr 587	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
32	31	anbi1d 630	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
33	27, 32	bitrd 278	1 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   class class class wbr 5074   × cxp 5587  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  0cc0 10871  ℝ^*cxr 11008   ≤ cle 11010   +_𝑒 cxad 12846  PsMetcpsmet 20581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-xr 11013  df-psmet 20589

This theorem is referenced by:  psmetdmdm  23458  psmetf  23459  psmet0  23461  psmettri2  23462  psmetres2  23467  xmetpsmet  23501