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Theorem rrxf 25248
Description: Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
rrxf.1 (𝜑𝐹𝑋)
Assertion
Ref Expression
rrxf (𝜑𝐹:𝐼⟶ℝ)
Distinct variable groups:   ,𝐹   ,𝐼
Allowed substitution hints:   𝜑()   𝑋()

Proof of Theorem rrxf
StepHypRef Expression
1 rrxmval.1 . . . 4 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
21ssrab3 4080 . . 3 𝑋 ⊆ (ℝ ↑m 𝐼)
3 rrxf.1 . . 3 (𝜑𝐹𝑋)
42, 3sselid 3980 . 2 (𝜑𝐹 ∈ (ℝ ↑m 𝐼))
5 elmapi 8849 . 2 (𝐹 ∈ (ℝ ↑m 𝐼) → 𝐹:𝐼⟶ℝ)
64, 5syl 17 1 (𝜑𝐹:𝐼⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {crab 3431   class class class wbr 5148  wf 6539  (class class class)co 7412  m cmap 8826   finSupp cfsupp 9367  cr 11115  0cc0 11116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828
This theorem is referenced by:  rrxsuppss  25250  rrxmval  25252  rrxmetlem  25254  rrxmet  25255  rrxdstprj1  25256
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