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| 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 2851 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) | 
| 4 | breq1 5146 | . . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0)) | |
| 5 | 4 | elrab 3692 | . . . 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 9409 | 1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 (class class class)co 7431 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 ℝcr 11154 0cc0 11155 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-fsupp 9402 | 
| This theorem is referenced by: rrxmval 25439 rrxmet 25442 rrxdstprj1 25443 | 
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