Theorem rrxfsupp 25444

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 2871	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0})
4		breq1 5102	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3650	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 220	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 499	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 9312	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413   class class class wbr 5099  (class class class)co 7392   supp csupp 8135   ↑_m cmap 8803  Fincfn 8923   finSupp cfsupp 9304  ℝcr 11069  0cc0 11070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-fsupp 9305

This theorem is referenced by:  rrxmval  25447  rrxmet  25450  rrxdstprj1  25451