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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 4080 | . . 3 ⊢ 𝑋 ⊆ (ℝ ↑m 𝐼) |
3 | rrxf.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑋) | |
4 | 2, 3 | sselid 3980 | . 2 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝐼)) |
5 | elmapi 8849 | . 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 2105 {crab 3431 class class class wbr 5148 ⟶wf 6539 (class class class)co 7412 ↑m cmap 8826 finSupp cfsupp 9367 ℝcr 11115 0cc0 11116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-map 8828 |
This theorem is referenced by: rrxsuppss 25250 rrxmval 25252 rrxmetlem 25254 rrxmet 25255 rrxdstprj1 25256 |
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