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Mirrors > Home > MPE Home > Th. List > coe1sfi | Structured version Visualization version GIF version |
Description: Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.) |
Ref | Expression |
---|---|
coe1sfi.a | ⊢ 𝐴 = (coe1‘𝐹) |
coe1sfi.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1sfi.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1sfi.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
coe1sfi | ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1sfi.a | . . 3 ⊢ 𝐴 = (coe1‘𝐹) | |
2 | coe1sfi.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | coe1sfi.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | df1o2 8411 | . . . 4 ⊢ 1o = {∅} | |
5 | nn0ex 12377 | . . . 4 ⊢ ℕ0 ∈ V | |
6 | 0ex 5262 | . . . 4 ⊢ ∅ ∈ V | |
7 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
8 | 4, 5, 6, 7 | mapsncnv 8789 | . . 3 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
9 | 1, 2, 3, 8 | coe1fval2 21533 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
10 | eqid 2737 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2737 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
12 | coe1sfi.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
13 | 3, 2 | ply1bascl2 21527 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
14 | 3, 2 | elbasfv 17049 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 10, 11, 12, 13, 14 | mplelsfi 21353 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
16 | 4, 5, 6, 7 | mapsnf1o2 8790 | . . . . 5 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 |
17 | f1ocnv 6793 | . . . . 5 ⊢ ((𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1-onto→(ℕ0 ↑m 1o)) | |
18 | f1of1 6780 | . . . . 5 ⊢ (◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1-onto→(ℕ0 ↑m 1o) → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) | |
19 | 16, 17, 18 | mp2b 10 | . . . 4 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o) |
20 | 19 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) |
21 | 12 | fvexi 6853 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 0 ∈ V) |
23 | id 22 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵) | |
24 | 15, 20, 22, 23 | fsuppco 9296 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) finSupp 0 ) |
25 | 9, 24 | eqbrtrd 5125 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 class class class wbr 5103 ↦ cmpt 5186 ◡ccnv 5630 ∘ ccom 5635 –1-1→wf1 6490 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7351 1oc1o 8397 ↑m cmap 8723 finSupp cfsupp 9263 ℕ0cn0 12371 Basecbs 17043 0gc0g 17281 mPoly cmpl 21261 Poly1cpl1 21500 coe1cco1 21501 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-tset 17112 df-ple 17113 df-psr 21264 df-mpl 21266 df-opsr 21268 df-psr1 21503 df-ply1 21505 df-coe1 21506 |
This theorem is referenced by: coe1fsupp 21537 mptcoe1fsupp 21538 ply1coefsupp 21618 mptcoe1matfsupp 22103 mp2pm2mplem4 22110 plypf1 25525 evls1fpws 32092 |
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