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

Theorem rrxfsupp 23932
Description: Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
rrxf.1 (𝜑𝐹𝑋)
Assertion
Ref Expression
rrxfsupp (𝜑 → (𝐹 supp 0) ∈ Fin)
Distinct variable groups:   ,𝐹   ,𝐼
Allowed substitution hints:   𝜑()   𝑋()

Proof of Theorem rrxfsupp
StepHypRef Expression
1 rrxf.1 . . . . 5 (𝜑𝐹𝑋)
2 rrxmval.1 . . . . 5 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
31, 2eleqtrdi 2920 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0})
4 breq1 5060 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3677 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 219 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
76simprd 496 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 8828 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139   class class class wbr 5057  (class class class)co 7145   supp csupp 7819  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821  cr 10524  0cc0 10525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-fsupp 8822
This theorem is referenced by:  rrxmval  23935  rrxmet  23938  rrxdstprj1  23939
  Copyright terms: Public domain W3C validator