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Theorem rrxfsupp 24004
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 2924 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0})
4 breq1 5045 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3655 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 221 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
76simprd 499 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 8828 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134   class class class wbr 5042  (class class class)co 7140   supp csupp 7817  m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  cr 10525  0cc0 10526
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-fsupp 8822
This theorem is referenced by:  rrxmval  24007  rrxmet  24010  rrxdstprj1  24011
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