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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 2843 | . . . 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 3683 | . . . 4 ⊢ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0)) |
6 | 3, 5 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐹 finSupp 0)) |
7 | 6 | simprd 496 | . 2 ⊢ (𝜑 → 𝐹 finSupp 0) |
8 | 7 | fsuppimpd 9371 | 1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 class class class wbr 5148 (class class class)co 7411 supp csupp 8148 ↑m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 ℝcr 11111 0cc0 11112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-fsupp 9364 |
This theorem is referenced by: rrxmval 24929 rrxmet 24932 rrxdstprj1 24933 |
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