Theorem ispsmet 22517

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 3414	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
2		id 22	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → 𝑢 = 𝑋)
3	2	sqxpeqd 5387	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
4	3	oveq2d 6938	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (ℝ* ↑𝑚 (𝑢 × 𝑢)) = (ℝ* ↑𝑚 (𝑋 × 𝑋)))
5		raleq 3330	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
6	5	raleqbi1dv 3328	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
7	6	anbi2d 622	. . . . . . . 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 3328	. . . . . . 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 3392	. . . . . 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 20134	. . . . . 6 ⊢ PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11		ovex 6954	. . . . . . 7 ⊢ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∈ V
12	11	rabex 5049	. . . . . 6 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
13	9, 10, 12	fvmpt 6542	. . . . 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 2845	. . 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 6928	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
17	16	eqeq1d 2780	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
18		oveq 6928	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
19		oveq 6928	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20		oveq 6928	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
21	19, 20	oveq12d 6940	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
22	18, 21	breq12d 4899	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
23	22	2ralbidv 3171	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
24	17, 23	anbi12d 624	. . . . 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 3168	. . . 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 3572	. . 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	syl6bb 279	. 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 12134	. . . 4 ⊢ ℝ* ∈ V
29		sqxpexg 7241	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
30		elmapg 8153	. . . 4 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
31	28, 29, 30	sylancr 581	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
32	31	anbi1d 623	. 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 271	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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  Vcvv 3398   class class class wbr 4886   × cxp 5353  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↑_𝑚 cmap 8140  0cc0 10272  ℝ^*cxr 10410   ≤ cle 10412   +_𝑒 cxad 12255  PsMetcpsmet 20126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-xr 10415  df-psmet 20134

This theorem is referenced by:  psmetdmdm  22518  psmetf  22519  psmet0  22521  psmettri2  22522  psmetres2  22527  xmetpsmet  22561