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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 2920 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) |
4 | breq1 5060 | . . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0)) | |
5 | 4 | elrab 3677 | . . . 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 8828 | 1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 (class class class)co 7145 supp csupp 7819 ↑m cmap 8395 Fincfn 8497 finSupp cfsupp 8821 ℝcr 10524 0cc0 10525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-fsupp 8822 |
This theorem is referenced by: rrxmval 23935 rrxmet 23938 rrxdstprj1 23939 |
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