Theorem rrxfsupp 23410

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	syl6eleq 2858	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0})
4		breq1 4786	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3512	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 208	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 483	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 8436	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1629   ∈ wcel 2143  {crab 3063   class class class wbr 4783  (class class class)co 6791   supp csupp 7444   ↑_𝑚 cmap 8007  Fincfn 8107   finSupp cfsupp 8429  ℝcr 10135  0cc0 10136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6794  df-fsupp 8430

This theorem is referenced by:  rrxmval  23413  rrxmet  23416  rrxdstprj1  23417