Theorem ispsmet 12973

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 12637	. . . . 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 4630	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑢 × 𝑢) = (𝑋 × 𝑋))
4	3	oveq2d 5858	. . . . . 6 ⊢ (𝑢 = 𝑋 → (ℝ* ↑𝑚 (𝑢 × 𝑢)) = (ℝ* ↑𝑚 (𝑋 × 𝑋)))
5		raleq 2661	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → (∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
6	5	raleqbi1dv 2669	. . . . . . . 8 ⊢ (𝑢 = 𝑋 → (∀𝑦 ∈ 𝑢 ∀𝑧 ∈ 𝑢 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
7	6	anbi2d 460	. . . . . . 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 2669	. . . . . 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 2721	. . . . 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 2737	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		xrex 9792	. . . . . . . 8 ⊢ ℝ* ∈ V
12		sqxpexg 4720	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V)
13		mapvalg 6624	. . . . . . . 8 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (ℝ* ↑𝑚 (𝑋 × 𝑋)) = {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*})
14	11, 12, 13	sylancr 411	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (ℝ* ↑𝑚 (𝑋 × 𝑋)) = {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*})
15		mapex 6620	. . . . . . . 8 ⊢ (((𝑋 × 𝑋) ∈ V ∧ ℝ* ∈ V) → {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*} ∈ V)
16	12, 11, 15	sylancl 410	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {𝑓 ∣ 𝑓:(𝑋 × 𝑋)⟶ℝ*} ∈ V)
17	14, 16	eqeltrd 2243	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (ℝ* ↑𝑚 (𝑋 × 𝑋)) ∈ V)
18		rabexg 4125	. . . . . 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 5579	. . . 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 2236	. . 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 5848	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑥) = (𝑥𝐷𝑥))
23	22	eqeq1d 2174	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑥) = 0 ↔ (𝑥𝐷𝑥) = 0))
24		oveq 5848	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
25		oveq 5848	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
26		oveq 5848	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
27	25, 26	oveq12d 5860	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
28	24, 27	breq12d 3995	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
29	28	2ralbidv 2490	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
30	23, 29	anbi12d 465	. . . . 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 2466	. . . 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 2882	. . 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 195	. 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 6627	. . . 4 ⊢ ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
35	11, 12, 34	sylancr 411	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (ℝ* ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
36	35	anbi1d 461	. 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 187	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 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  {crab 2448  Vcvv 2726   class class class wbr 3982   × cxp 4602  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842   ↑_𝑚 cmap 6614  0cc0 7753  ℝ^*cxr 7932   ≤ cle 7934   +_𝑒 cxad 9706  PsMetcpsmet 12629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-psmet 12637

This theorem is referenced by:  psmetdmdm  12974  psmetf  12975  psmet0  12977  psmettri2  12978  psmetres2  12983  xmetpsmet  13019