Theorem metuval 22573

Description: Value of the uniform structure generated by metric 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
metuval	⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))

Distinct variable groups:   𝐷,𝑎   𝑋,𝑎

Proof of Theorem metuval

Dummy variables 𝑢 𝑑 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-metu 19959	. . 3 ⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
2	1	a1i 11	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))))))
3		simpr 471	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
4	3	dmeqd 5464	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
5	4	dmeqd 5464	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
6		psmetdmdm 22329	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
7	6	adantr 466	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
8	5, 7	eqtr4d 2808	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
9	8	sqxpeqd 5281	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 × dom dom 𝑑) = (𝑋 × 𝑋))
10		simplr 744	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
11	10	cnveqd 5436	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → ◡𝑑 = ◡𝐷)
12	11	imaeq1d 5606	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → (◡𝑑 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑎)))
13	12	mpteq2dva 4878	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
14	13	rneqd 5491	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
15	9, 14	oveq12d 6810	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))
16		elfvdm 6361	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
17		fveq2 6332	. . . . . 6 ⊢ (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
18	17	eleq2d 2836	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
19	18	rspcev 3460	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
20	16, 19	mpancom 660	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
21		df-psmet 19952	. . . . 5 ⊢ PsMet = (𝑦 ∈ V ↦ {𝑢 ∈ (ℝ* ↑𝑚 (𝑦 × 𝑦)) ∣ ∀𝑧 ∈ 𝑦 ((𝑧𝑢𝑧) = 0 ∧ ∀𝑤 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑧𝑢𝑤) ≤ ((𝑣𝑢𝑧) +𝑒 (𝑣𝑢𝑤)))})
22	21	funmpt2 6070	. . . 4 ⊢ Fun PsMet
23		elunirn 6651	. . . 4 ⊢ (Fun PsMet → (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
24	22, 23	ax-mp 5	. . 3 ⊢ (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
25	20, 24	sylibr 224	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
26		ovexd 6824	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∈ V)
27	2, 15, 25, 26	fvmptd 6430	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351  ∪ cuni 4574   class class class wbr 4786   ↦ cmpt 4863   × cxp 5247  ^◡ccnv 5248  dom cdm 5249  ran crn 5250   “ cima 5252  Fun wfun 6025  ‘cfv 6031  (class class class)co 6792   ↑_𝑚 cmap 8008  0cc0 10137  ℝ^*cxr 10274   ≤ cle 10276  ℝ⁺crp 12034   +_𝑒 cxad 12148  [,)cico 12381  PsMetcpsmet 19944  filGencfg 19949  metUnifcmetu 19951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-xr 10279  df-psmet 19952  df-metu 19959

This theorem is referenced by:  metuust  22584  cfilucfil2  22585  metuel  22588  psmetutop  22591  restmetu  22594  metucn  22595