Theorem rrxfsupp 25416

Description: Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.)

Hypotheses

Ref	Expression
rrxmval.1	⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}
rrxf.1	⊢ (𝜑 → 𝐹 ∈ 𝑋)

Assertion

Ref	Expression
rrxfsupp	⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Distinct variable groups:   ℎ,𝐹   ℎ,𝐼

Allowed substitution hints:   𝜑(ℎ)   𝑋(ℎ)

Proof of Theorem rrxfsupp

Step	Hyp	Ref	Expression
1		rrxf.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
2		rrxmval.1	. . . . 5 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}
3	1, 2	eleqtrdi 2836	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0})
4		breq1 5147	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3681	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 217	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 494	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 9404	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {crab 3420   class class class wbr 5144  (class class class)co 7414   supp csupp 8164   ↑_m cmap 8845  Fincfn 8964   finSupp cfsupp 9396  ℝcr 11146  0cc0 11147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7417  df-fsupp 9397

This theorem is referenced by:  rrxmval  25419  rrxmet  25422  rrxdstprj1  25423