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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 2848 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) |
4 | breq1 5109 | . . . . 5 ⊢ (ℎ = 𝐹 → (ℎ finSupp 0 ↔ 𝐹 finSupp 0)) | |
5 | 4 | elrab 3646 | . . . 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 497 | . 2 ⊢ (𝜑 → 𝐹 finSupp 0) |
8 | 7 | fsuppimpd 9313 | 1 ⊢ (𝜑 → (𝐹 supp 0) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3408 class class class wbr 5106 (class class class)co 7358 supp csupp 8093 ↑m cmap 8766 Fincfn 8884 finSupp cfsupp 9306 ℝcr 11051 0cc0 11052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-fsupp 9307 |
This theorem is referenced by: rrxmval 24772 rrxmet 24775 rrxdstprj1 24776 |
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