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| Mirrors > Home > MPE Home > Th. List > rrxf | Structured version Visualization version GIF version | ||
| Description: Euclidean vectors as functions. (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 | 
|---|---|
| rrxf | ⊢ (𝜑 → 𝐹:𝐼⟶ℝ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rrxmval.1 | . . . 4 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} | |
| 2 | 1 | ssrab3 4082 | . . 3 ⊢ 𝑋 ⊆ (ℝ ↑m 𝐼) | 
| 3 | rrxf.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑋) | |
| 4 | 2, 3 | sselid 3981 | . 2 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝐼)) | 
| 5 | elmapi 8889 | . 2 ⊢ (𝐹 ∈ (ℝ ↑m 𝐼) → 𝐹:𝐼⟶ℝ) | |
| 6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → 𝐹:𝐼⟶ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 | 
| This theorem is referenced by: rrxsuppss 25437 rrxmval 25439 rrxmetlem 25441 rrxmet 25442 rrxdstprj1 25443 | 
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