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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 8513 | . . . 4 ⊢ 1o = {∅} | |
| 5 | nn0ex 12532 | . . . 4 ⊢ ℕ0 ∈ V | |
| 6 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 7 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
| 8 | 4, 5, 6, 7 | mapsncnv 8933 | . . 3 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
| 9 | 1, 2, 3, 8 | coe1fval2 22212 | . 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 22206 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
| 14 | 10, 11, 12, 13 | mplelsfi 22015 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
| 15 | 4, 5, 6, 7 | mapsnf1o2 8934 | . . . . 5 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 |
| 16 | f1ocnv 6860 | . . . . 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)) | |
| 17 | f1of1 6847 | . . . . 5 ⊢ (◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1-onto→(ℕ0 ↑m 1o) → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) | |
| 18 | 15, 16, 17 | mp2b 10 | . . . 4 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o) |
| 19 | 18 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) |
| 20 | 12 | fvexi 6920 | . . . 4 ⊢ 0 ∈ V |
| 21 | 20 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 0 ∈ V) |
| 22 | id 22 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵) | |
| 23 | 14, 19, 21, 22 | fsuppco 9442 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) finSupp 0 ) |
| 24 | 9, 23 | eqbrtrd 5165 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 ∘ ccom 5689 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ↑m cmap 8866 finSupp cfsupp 9401 ℕ0cn0 12526 Basecbs 17247 0gc0g 17484 mPoly cmpl 21926 Poly1cpl1 22178 coe1cco1 22179 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-psr 21929 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-ply1 22183 df-coe1 22184 |
| This theorem is referenced by: coe1fsupp 22216 mptcoe1fsupp 22217 ply1coefsupp 22301 evls1fpws 22373 mptcoe1matfsupp 22808 mp2pm2mplem4 22815 plypf1 26251 |
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