Theorem ispsmet 15040

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 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psmet 14550	. . . . 5 ⊢ PsMet = (𝑢 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑢 × 𝑢)) ∣ ∀𝑥 ∈ 𝑢 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
2		id 19	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → 𝑢 = 𝑋)
3	2	sqxpeqd 4749	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
4	3	oveq2d 6029	. . . . . 6 ⊢ (𝑢 = 𝑋 → (ℝ* ↑𝑚 (𝑢 × 𝑢)) = (ℝ* ↑𝑚 (𝑋 × 𝑋)))
5		raleq 2728	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
6	5	raleqbi1dv 2740	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
7	6	anbi2d 464	. . . . . . 7 ⊢ (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 2740	. . . . . 6 ⊢ (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 2795	. . . . 5 ⊢ (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		elex 2812	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		xrex 10084	. . . . . . . 8 ⊢ ℝ* ∈ V
12		sqxpexg 4841	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
13		mapvalg 6822	. . . . . . . 8 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (ℝ* ↑𝑚 (𝑋 × 𝑋)) = {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*})
14	11, 12, 13	sylancr 414	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (ℝ* ↑𝑚 (𝑋 × 𝑋)) = {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*})
15		mapex 6818	. . . . . . . 8 ⊢ (((𝑋 × 𝑋) ∈ V ∧ ℝ* ∈ V) → {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*} ∈ V)
16	12, 11, 15	sylancl 413	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*} ∈ V)
17	14, 16	eqeltrd 2306	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∈ V)
18		rabexg 4231	. . . . . 6 ⊢ ((ℝ* ↑𝑚 (𝑋 × 𝑋)) ∈ V → {𝑑 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V)
19	17, 18	syl 14	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → {𝑑 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V)
20	1, 9, 10, 19	fvmptd3 5736	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (PsMet‘𝑋) = {𝑑 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
21	20	eleq2d 2299	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
22		oveq 6019	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
23	22	eqeq1d 2238	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
24		oveq 6019	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
25		oveq 6019	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
26		oveq 6019	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
27	25, 26	oveq12d 6031	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
28	24, 27	breq12d 4099	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
29	28	2ralbidv 2554	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
30	23, 29	anbi12d 473	. . . . 5 ⊢ (𝑑 = 𝐷 → (((𝑥𝑑𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
31	30	ralbidv 2530	. . . 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)))))
32	31	elrab 2960	. . 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)))))
33	21, 32	bitrdi 196	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
34		elmapg 6825	. . . 4 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
35	11, 12, 34	sylancr 414	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
36	35	anbi1d 465	. 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))))))
37	33, 36	bitrd 188	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   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  {crab 2512  Vcvv 2800   class class class wbr 4086   × cxp 4721  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013   ↑_𝑚 cmap 6812  0cc0 8025  ℝ^*cxr 8206   ≤ cle 8208   +_𝑒 cxad 9998  PsMetcpsmet 14542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-map 6814  df-pnf 8209  df-mnf 8210  df-xr 8211  df-psmet 14550

This theorem is referenced by:  psmetdmdm  15041  psmetf  15042  psmet0  15044  psmettri2  15045  psmetres2  15050  xmetpsmet  15086