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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 5144 | . 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 22085 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) |
7 | 5 | fvexi 6899 | . . . . 5 ⊢ 𝐾 ∈ V |
8 | nn0ex 12482 | . . . . 5 ⊢ ℕ0 ∈ V | |
9 | 7, 8 | pm3.2i 470 | . . . 4 ⊢ (𝐾 ∈ V ∧ ℕ0 ∈ V) |
10 | elmapg 8835 | . . . 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 257 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ (𝐾 ↑m ℕ0)) |
13 | coe1sfi.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
14 | 2, 3, 4, 13 | coe1sfi 22087 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
15 | 1, 12, 14 | elrabd 3680 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m ℕ0) ∣ 𝑔 finSupp 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 class class class wbr 5141 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 finSupp cfsupp 9363 ℕ0cn0 12476 Basecbs 17153 0gc0g 17394 Poly1cpl1 22051 coe1cco1 22052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-tset 17225 df-ple 17226 df-psr 21803 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-ply1 22056 df-coe1 22057 |
This theorem is referenced by: mptcoe1fsupp 22089 coe1ae0 22090 pmatcoe1fsupp 22558 mptcoe1matfsupp 22659 mp2pm2mplem4 22666 |
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