![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rrxsuppss | Structured version Visualization version GIF version |
Description: Support of Euclidean vectors. (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 |
---|---|
rrxsuppss | ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssdm 8108 | . 2 ⊢ (𝐹 supp 0) ⊆ dom 𝐹 | |
2 | rrxmval.1 | . . 3 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} | |
3 | rrxf.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑋) | |
4 | 2, 3 | rrxf 24765 | . 2 ⊢ (𝜑 → 𝐹:𝐼⟶ℝ) |
5 | 1, 4 | fssdm 6688 | 1 ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3407 ⊆ wss 3910 class class class wbr 5105 (class class class)co 7357 supp csupp 8092 ↑m cmap 8765 finSupp cfsupp 9305 ℝcr 11050 0cc0 11051 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
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-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-supp 8093 df-map 8767 |
This theorem is referenced by: rrxmval 24769 rrxmet 24772 rrxdstprj1 24773 |
Copyright terms: Public domain | W3C validator |