Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | coe1sfi.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
2 | coe1sfi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
3 | coe1sfi.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1fvalcl.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | 1, 2, 3, 4 | coe1f 19402 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) |
6 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
7 | 4, 6 | eqeltri 2684 | . . . . 5 ⊢ 𝐾 ∈ V |
8 | nn0ex 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
9 | 7, 8 | pm3.2i 470 | . . . 4 ⊢ (𝐾 ∈ V ∧ ℕ0 ∈ V) |
10 | elmapg 7757 | . . . 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 | 5, 11 | mpbird 246 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ (𝐾 ↑𝑚 ℕ0)) |
13 | coe1sfi.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
14 | 1, 2, 3, 13 | coe1sfi 19404 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
15 | breq1 4586 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑔 finSupp 0 ↔ 𝐴 finSupp 0 )) | |
16 | 15 | elrab 3331 | . 2 ⊢ (𝐴 ∈ {𝑔 ∈ (𝐾 ↑𝑚 ℕ0) ∣ 𝑔 finSupp 0 } ↔ (𝐴 ∈ (𝐾 ↑𝑚 ℕ0) ∧ 𝐴 finSupp 0 )) |
17 | 12, 14, 16 | sylanbrc 695 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑𝑚 ℕ0) ∣ 𝑔 finSupp 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 finSupp cfsupp 8158 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 Poly1cpl1 19368 coe1cco1 19369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 df-coe1 19374 |
This theorem is referenced by: mptcoe1fsupp 19406 coe1ae0 19407 pmatcoe1fsupp 20325 mptcoe1matfsupp 20426 mp2pm2mplem4 20433 |
Copyright terms: Public domain | W3C validator |