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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 2823 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
4 | eqid 2823 | . . . 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 20212 | . . 3 ⊢ (𝐹 ∈ 𝐵 ↔ (𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝐹 finSupp 0 )) |
8 | 7 | simprbi 499 | . 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 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 finSupp cfsupp 8835 Basecbs 16485 0gc0g 16715 mPwSer cmps 20133 mPoly cmpl 20135 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-psr 20138 df-mpl 20140 |
This theorem is referenced by: evlslem2 20294 evlslem6 20296 coe1sfi 20383 mdegldg 24662 mdegcl 24665 selvval2lem4 39143 |
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