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Theorem rrxsuppss 24300
Description: Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
rrxf.1 (𝜑𝐹𝑋)
Assertion
Ref Expression
rrxsuppss (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
Distinct variable groups:   ,𝐹   ,𝐼
Allowed substitution hints:   𝜑()   𝑋()

Proof of Theorem rrxsuppss
StepHypRef Expression
1 suppssdm 7919 . 2 (𝐹 supp 0) ⊆ dom 𝐹
2 rrxmval.1 . . 3 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
3 rrxf.1 . . 3 (𝜑𝐹𝑋)
42, 3rrxf 24298 . 2 (𝜑𝐹:𝐼⟶ℝ)
51, 4fssdm 6565 1 (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {crab 3065  wss 3866   class class class wbr 5053  (class class class)co 7213   supp csupp 7903  m cmap 8508   finSupp cfsupp 8985  cr 10728  0cc0 10729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-supp 7904  df-map 8510
This theorem is referenced by:  rrxmval  24302  rrxmet  24305  rrxdstprj1  24306
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