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Mirrors > Home > MPE Home > Th. List > mplelsfi | Structured version Visualization version GIF version |
Description: A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
Ref | Expression |
---|---|
mplrcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mplelsfi.z | ⊢ 0 = (0g‘𝑅) |
mplelsfi.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mplelsfi.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
Ref | Expression |
---|---|
mplelsfi | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplelsfi.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
2 | mplrcl.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | eqid 2777 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
4 | eqid 2777 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
5 | mplelsfi.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
6 | mplrcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 2, 3, 4, 5, 6 | mplelbas 19827 | . . 3 ⊢ (𝐹 ∈ 𝐵 ↔ (𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝐹 finSupp 0 )) |
8 | 7 | simprbi 492 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
9 | 1, 8 | syl 17 | 1 ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 finSupp cfsupp 8563 Basecbs 16255 0gc0g 16486 mPwSer cmps 19748 mPoly cmpl 19750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-1cn 10330 ax-addcl 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-nn 11375 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-psr 19753 df-mpl 19755 |
This theorem is referenced by: evlslem2 19908 evlslem6 19909 coe1sfi 19979 mdegldg 24263 mdegcl 24266 |
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