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Mirrors > Home > MPE Home > Th. List > coe1fsupp | Structured version Visualization version GIF version |
Description: The coefficient vector of a univariate polynomial is a finitely supported mapping from the nonnegative integers to the elements of the coefficient class/ring for the polynomial. (Contributed by AV, 3-Oct-2019.) |
Ref | Expression |
---|---|
coe1sfi.a | ⊢ 𝐴 = (coe1‘𝐹) |
coe1sfi.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1sfi.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1sfi.z | ⊢ 0 = (0g‘𝑅) |
coe1fvalcl.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
coe1fsupp | ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m ℕ0) ∣ 𝑔 finSupp 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5034 | . 2 ⊢ (𝑔 = 𝐴 → (𝑔 finSupp 0 ↔ 𝐴 finSupp 0 )) | |
2 | coe1sfi.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
3 | coe1sfi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | coe1sfi.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | coe1fvalcl.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | coe1f 20989 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) |
7 | 5 | fvexi 6691 | . . . . 5 ⊢ 𝐾 ∈ V |
8 | nn0ex 11985 | . . . . 5 ⊢ ℕ0 ∈ V | |
9 | 7, 8 | pm3.2i 474 | . . . 4 ⊢ (𝐾 ∈ V ∧ ℕ0 ∈ V) |
10 | elmapg 8453 | . . . 4 ⊢ ((𝐾 ∈ V ∧ ℕ0 ∈ V) → (𝐴 ∈ (𝐾 ↑m ℕ0) ↔ 𝐴:ℕ0⟶𝐾)) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐴 ∈ (𝐾 ↑m ℕ0) ↔ 𝐴:ℕ0⟶𝐾)) |
12 | 6, 11 | mpbird 260 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ (𝐾 ↑m ℕ0)) |
13 | coe1sfi.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
14 | 2, 3, 4, 13 | coe1sfi 20991 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
15 | 1, 12, 14 | elrabd 3591 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m ℕ0) ∣ 𝑔 finSupp 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3058 Vcvv 3399 class class class wbr 5031 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ↑m cmap 8440 finSupp cfsupp 8909 ℕ0cn0 11979 Basecbs 16589 0gc0g 16819 Poly1cpl1 20955 coe1cco1 20956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-tset 16690 df-ple 16691 df-psr 20725 df-mpl 20727 df-opsr 20729 df-psr1 20958 df-ply1 20960 df-coe1 20961 |
This theorem is referenced by: mptcoe1fsupp 20993 coe1ae0 20994 pmatcoe1fsupp 21455 mptcoe1matfsupp 21556 mp2pm2mplem4 21563 |
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