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Theorem rrxfsupp 22993
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, 2syl6eleq 2698 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0})
4 breq1 4586 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3331 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 207 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
76simprd 478 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 8165 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900   class class class wbr 4583  (class class class)co 6549   supp csupp 7182  𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  cr 9814  0cc0 9815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-fsupp 8159
This theorem is referenced by:  rrxmval  22996  rrxmet  22999  rrxdstprj1  23000
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