Theorem rrxfsupp 23608

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 2869	. . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0})
4		breq1 4889	. . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0))
5	4	elrab 3572	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
6	3, 5	sylib 210	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝐼) ∧ 𝐹 finSupp 0))
7	6	simprd 491	. 2 ⊢ (𝜑 → 𝐹 finSupp 0)
8	7	fsuppimpd 8570	1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {crab 3094   class class class wbr 4886  (class class class)co 6922   supp csupp 7576   ↑_𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  ℝcr 10271  0cc0 10272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-fsupp 8564

This theorem is referenced by:  rrxmval  23611  rrxmet  23614  rrxdstprj1  23615