Theorem rrxfsupp 25360

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 2845	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0})
4		breq1 5100	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3645	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 218	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 495	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 9274	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3398   class class class wbr 5097  (class class class)co 7358   supp csupp 8102   ↑_m cmap 8765  Fincfn 8885   finSupp cfsupp 9266  ℝcr 11027  0cc0 11028

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-iota 6447  df-fun 6493  df-fv 6499  df-ov 7361  df-fsupp 9267

This theorem is referenced by:  rrxmval  25363  rrxmet  25366  rrxdstprj1  25367