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Theorem rrxsuppss 24017
 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 7829 . 2 (𝐹 supp 0) ⊆ dom 𝐹
2 rrxmval.1 . . 3 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
3 rrxf.1 . . 3 (𝜑𝐹𝑋)
42, 3rrxf 24015 . 2 (𝜑𝐹:𝐼⟶ℝ)
51, 4fssdm 6505 1 (𝜑 → (𝐹 supp 0) ⊆ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110   ⊆ wss 3881   class class class wbr 5031  (class class class)co 7136   supp csupp 7816   ↑m cmap 8392   finSupp cfsupp 8820  ℝcr 10528  0cc0 10529 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-supp 7817  df-map 8394 This theorem is referenced by:  rrxmval  24019  rrxmet  24022  rrxdstprj1  24023
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