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Theorem rrxfsupp 24926
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 2843 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0})
4 breq1 5151 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3683 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 217 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
76simprd 496 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 9371 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3432   class class class wbr 5148  (class class class)co 7411   supp csupp 8148  m cmap 8822  Fincfn 8941   finSupp cfsupp 9363  cr 11111  0cc0 11112
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-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-fsupp 9364
This theorem is referenced by:  rrxmval  24929  rrxmet  24932  rrxdstprj1  24933
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