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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 8099 | . . . 4 ⊢ 1o = {∅} | |
5 | nn0ex 11891 | . . . 4 ⊢ ℕ0 ∈ V | |
6 | 0ex 5175 | . . . 4 ⊢ ∅ ∈ V | |
7 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
8 | 4, 5, 6, 7 | mapsncnv 8440 | . . 3 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
9 | 1, 2, 3, 8 | coe1fval2 20839 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
10 | eqid 2798 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2798 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
12 | coe1sfi.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
13 | 3, 2 | ply1bascl2 20833 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
14 | 3, 2 | elbasfv 16536 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 10, 11, 12, 13, 14 | mplelsfi 20730 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
16 | 4, 5, 6, 7 | mapsnf1o2 8441 | . . . . 5 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 |
17 | f1ocnv 6602 | . . . . 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 6589 | . . . . 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 6659 | . . . 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 8849 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) finSupp 0 ) |
25 | 9, 24 | eqbrtrd 5052 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 ∘ ccom 5523 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 finSupp cfsupp 8817 ℕ0cn0 11885 Basecbs 16475 0gc0g 16705 mPoly cmpl 20591 Poly1cpl1 20806 coe1cco1 20807 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 df-coe1 20812 |
This theorem is referenced by: coe1fsupp 20843 mptcoe1fsupp 20844 ply1coefsupp 20924 mptcoe1matfsupp 21407 mp2pm2mplem4 21414 plypf1 24809 |
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