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Theorem rrxfsupp 23608
 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 2869 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0})
4 breq1 4889 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3572 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 210 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
76simprd 491 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 8570 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {crab 3094   class class class wbr 4886  (class class class)co 6922   supp csupp 7576   ↑𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  ℝcr 10271  0cc0 10272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-fsupp 8564 This theorem is referenced by:  rrxmval  23611  rrxmet  23614  rrxdstprj1  23615
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