Theorem ispsmet 24603

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 3472	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
2		id 23	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → 𝑢 = 𝑋)
3	2	sqxpeqd 5683	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
4	3	oveq2d 7428	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (ℝ* ↑m (𝑢 × 𝑢)) = (ℝ* ↑m (𝑋 × 𝑋)))
5		raleq 3317	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
6	5	raleqbi1dv 3330	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
7	6	anbi2d 642	. . . . . . . 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 3330	. . . . . . 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 3430	. . . . . 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 21650	. . . . . 6 ⊢ PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑢 × 𝑢)) ∣ ∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11		ovex 7445	. . . . . . 7 ⊢ (ℝ* ↑m (𝑋 × 𝑋)) ∈ V
12	11	rabex 5300	. . . . . 6 ⊢ {𝑑 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
13	9, 10, 12	fvmpt 6985	. . . . 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 18	. . . 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 2847	. . 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 7418	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
17	16	eqeq1d 2763	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18		oveq 7418	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19		oveq 7418	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20		oveq 7418	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
21	19, 20	oveq12d 7430	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
22	18, 21	breq12d 5116	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23	22	2ralbidv 3227	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
24	17, 23	anbi12d 644	. . . . 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 3186	. . . 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 3645	. . 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 290	. 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 13096	. . . 4 ⊢ ℝ* ∈ V
29		sqxpexg 7758	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
30		elmapg 8843	. . . 4 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
31	28, 29, 30	sylancr 599	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (ℝ* ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
32	31	anbi1d 643	. 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 282	1 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451   class class class wbr 5103   × cxp 5649  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  0cc0 11181  ℝ^*cxr 11323   ≤ cle 11325   +_𝑒 cxad 13220  PsMetcpsmet 21642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8833  df-xr 11328  df-psmet 21650

This theorem is used by:  psmetdmdm  24604  psmetf  24605  psmet0  24607  psmettri2  24608  psmetres2  24613  xmetpsmet  24647