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Mirrors > Home > MPE Home > Th. List > coe1ae0 | Structured version Visualization version GIF version |
Description: The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.) |
Ref | Expression |
---|---|
coe1ae0.a | ⊢ 𝐴 = (coe1‘𝐹) |
coe1ae0.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1ae0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1ae0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
coe1ae0 | ⊢ (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1ae0.a | . . 3 ⊢ 𝐴 = (coe1‘𝐹) | |
2 | coe1ae0.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | coe1ae0.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1ae0.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | eqid 2732 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 1, 2, 3, 4, 5 | coe1fsupp 21737 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑎 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑎 finSupp 0 }) |
7 | breq1 5151 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 finSupp 0 ↔ 𝐴 finSupp 0 )) | |
8 | 7 | elrab 3683 | . . 3 ⊢ (𝐴 ∈ {𝑎 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑎 finSupp 0 } ↔ (𝐴 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝐴 finSupp 0 )) |
9 | 4 | fvexi 6905 | . . . . . 6 ⊢ 0 ∈ V |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 0 ∈ V) |
11 | fsuppmapnn0ub 13959 | . . . . 5 ⊢ ((𝐴 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 0 ∈ V) → (𝐴 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 ))) | |
12 | 10, 11 | sylan2 593 | . . . 4 ⊢ ((𝐴 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝐹 ∈ 𝐵) → (𝐴 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 ))) |
13 | 12 | impancom 452 | . . 3 ⊢ ((𝐴 ∈ ((Base‘𝑅) ↑m ℕ0) ∧ 𝐴 finSupp 0 ) → (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 ))) |
14 | 8, 13 | sylbi 216 | . 2 ⊢ (𝐴 ∈ {𝑎 ∈ ((Base‘𝑅) ↑m ℕ0) ∣ 𝑎 finSupp 0 } → (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 ))) |
15 | 6, 14 | mpcom 38 | 1 ⊢ (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ↑m cmap 8819 finSupp cfsupp 9360 < clt 11247 ℕ0cn0 12471 Basecbs 17143 0gc0g 17384 Poly1cpl1 21700 coe1cco1 21701 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-tset 17215 df-ple 17216 df-psr 21461 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-ply1 21705 df-coe1 21706 |
This theorem is referenced by: pmatcollpw1lem1 22275 evls1fpws 32641 ply1mulgsumlem1 47057 |
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