MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caufval Structured version   Visualization version   GIF version

Theorem caufval 24791
Description: The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)
Assertion
Ref Expression
caufval (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (Cauβ€˜π·) = {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)})
Distinct variable groups:   𝑓,π‘˜,π‘₯,𝐷   𝑓,𝑋,π‘˜,π‘₯

Proof of Theorem caufval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-cau 24772 . 2 Cau = (𝑑 ∈ βˆͺ ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯)})
2 dmeq 5903 . . . . . 6 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
32dmeqd 5905 . . . . 5 (𝑑 = 𝐷 β†’ dom dom 𝑑 = dom dom 𝐷)
4 xmetf 23834 . . . . . . . 8 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„*)
54fdmd 6728 . . . . . . 7 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ dom 𝐷 = (𝑋 Γ— 𝑋))
65dmeqd 5905 . . . . . 6 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ dom dom 𝐷 = dom (𝑋 Γ— 𝑋))
7 dmxpid 5929 . . . . . 6 dom (𝑋 Γ— 𝑋) = 𝑋
86, 7eqtrdi 2788 . . . . 5 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ dom dom 𝐷 = 𝑋)
93, 8sylan9eqr 2794 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = 𝑋)
109oveq1d 7423 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (dom dom 𝑑 ↑pm β„‚) = (𝑋 ↑pm β„‚))
11 simpr 485 . . . . . . . 8 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
1211fveq2d 6895 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (ballβ€˜π‘‘) = (ballβ€˜π·))
1312oveqd 7425 . . . . . 6 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯) = ((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯))
1413feq3d 6704 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯) ↔ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)))
1514rexbidv 3178 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯) ↔ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)))
1615ralbidv 3177 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)))
1710, 16rabeqbidv 3449 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ {𝑓 ∈ (dom dom 𝑑 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π‘‘)π‘₯)} = {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)})
18 fvssunirn 6924 . . 3 (∞Metβ€˜π‘‹) βŠ† βˆͺ ran ∞Met
1918sseli 3978 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐷 ∈ βˆͺ ran ∞Met)
20 ovex 7441 . . . 4 (𝑋 ↑pm β„‚) ∈ V
2120rabex 5332 . . 3 {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)} ∈ V
2221a1i 11 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)} ∈ V)
231, 17, 19, 22fvmptd2 7006 1 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (Cauβ€˜π·) = {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474  βˆͺ cuni 4908   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑pm cpm 8820  β„‚cc 11107  β„*cxr 11246  β„€cz 12557  β„€β‰₯cuz 12821  β„+crp 12973  βˆžMetcxmet 20928  ballcbl 20930  Cauccau 24769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-xr 11251  df-xmet 20936  df-cau 24772
This theorem is referenced by:  iscau  24792  equivcau  24816
  Copyright terms: Public domain W3C validator