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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 8119 | . . . 4 ⊢ 1o = {∅} | |
5 | nn0ex 11906 | . . . 4 ⊢ ℕ0 ∈ V | |
6 | 0ex 5214 | . . . 4 ⊢ ∅ ∈ V | |
7 | eqid 2824 | . . . 4 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
8 | 4, 5, 6, 7 | mapsncnv 8460 | . . 3 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
9 | 1, 2, 3, 8 | coe1fval2 20381 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
10 | eqid 2824 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2824 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
12 | coe1sfi.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
13 | 3, 2 | ply1bascl2 20375 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
14 | 3, 2 | elbasfv 16547 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V) |
15 | 10, 11, 12, 13, 14 | mplelsfi 20274 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
16 | 4, 5, 6, 7 | mapsnf1o2 8461 | . . . . 5 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 |
17 | f1ocnv 6630 | . . . . 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 6617 | . . . . 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 6687 | . . . 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 8868 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) finSupp 0 ) |
25 | 9, 24 | eqbrtrd 5091 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 class class class wbr 5069 ↦ cmpt 5149 ◡ccnv 5557 ∘ ccom 5562 –1-1→wf1 6355 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 ↑m cmap 8409 finSupp cfsupp 8836 ℕ0cn0 11900 Basecbs 16486 0gc0g 16716 mPoly cmpl 20136 Poly1cpl1 20348 coe1cco1 20349 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-psr 20139 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-ply1 20353 df-coe1 20354 |
This theorem is referenced by: coe1fsupp 20385 mptcoe1fsupp 20386 ply1coefsupp 20466 mptcoe1matfsupp 21413 mp2pm2mplem4 21420 plypf1 24805 |
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