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Mirrors > Home > MPE Home > Th. List > rrxfsupp | Structured version Visualization version GIF version |
Description: Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
Ref | Expression |
---|---|
rrxmval.1 | ⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} |
rrxf.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑋) |
Ref | Expression |
---|---|
rrxfsupp | ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) |
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 2849 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) |
4 | breq1 5151 | . . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0)) | |
5 | 4 | elrab 3695 | . . . 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 9407 | 1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 (class class class)co 7431 supp csupp 8184 ↑m cmap 8865 Fincfn 8984 finSupp cfsupp 9399 ℝcr 11152 0cc0 11153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-fsupp 9400 |
This theorem is referenced by: rrxmval 25453 rrxmet 25456 rrxdstprj1 25457 |
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