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Theorem rrxfsupp 24919
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 2844 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0})
4 breq1 5152 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3684 . . . 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 497 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 9369 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3433   class class class wbr 5149  (class class class)co 7409   supp csupp 8146  m cmap 8820  Fincfn 8939   finSupp cfsupp 9361  cr 11109  0cc0 11110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-fsupp 9362
This theorem is referenced by:  rrxmval  24922  rrxmet  24925  rrxdstprj1  24926
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