Theorem rrxfsupp 25394

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 2850	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0})
4		breq1 5082	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3636	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 219	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 496	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 9279	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392   class class class wbr 5079  (class class class)co 7363   supp csupp 8107   ↑_m cmap 8770  Fincfn 8890   finSupp cfsupp 9271  ℝcr 11035  0cc0 11036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-fsupp 9272

This theorem is referenced by:  rrxmval  25397  rrxmet  25400  rrxdstprj1  25401