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Theorem rrxfsupp 23410
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 2858 . . . 4 (𝜑𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0})
4 breq1 4786 . . . . 5 ( = 𝐹 → ( finSupp 0 ↔ 𝐹 finSupp 0))
54elrab 3512 . . . 4 (𝐹 ∈ { ∈ (ℝ ↑𝑚 𝐼) ∣ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
63, 5sylib 208 . . 3 (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
76simprd 483 . 2 (𝜑𝐹 finSupp 0)
87fsuppimpd 8436 1 (𝜑 → (𝐹 supp 0) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wcel 2143  {crab 3063   class class class wbr 4783  (class class class)co 6791   supp csupp 7444  𝑚 cmap 8007  Fincfn 8107   finSupp cfsupp 8429  cr 10135  0cc0 10136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6794  df-fsupp 8430
This theorem is referenced by:  rrxmval  23413  rrxmet  23416  rrxdstprj1  23417
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