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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 8512 | . . . 4 ⊢ 1o = {∅} | |
5 | nn0ex 12530 | . . . 4 ⊢ ℕ0 ∈ V | |
6 | 0ex 5313 | . . . 4 ⊢ ∅ ∈ V | |
7 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) | |
8 | 4, 5, 6, 7 | mapsncnv 8932 | . . 3 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)) = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) |
9 | 1, 2, 3, 8 | coe1fval2 22228 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)))) |
10 | eqid 2735 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
11 | eqid 2735 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
12 | coe1sfi.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
13 | 3, 2 | ply1bascl2 22222 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
14 | 10, 11, 12, 13 | mplelsfi 22033 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 finSupp 0 ) |
15 | 4, 5, 6, 7 | mapsnf1o2 8933 | . . . . 5 ⊢ (𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0 |
16 | f1ocnv 6861 | . . . . 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)) | |
17 | f1of1 6848 | . . . . 5 ⊢ (◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1-onto→(ℕ0 ↑m 1o) → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) | |
18 | 15, 16, 17 | mp2b 10 | . . . 4 ⊢ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o) |
19 | 18 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅)):ℕ0–1-1→(ℕ0 ↑m 1o)) |
20 | 12 | fvexi 6921 | . . . 4 ⊢ 0 ∈ V |
21 | 20 | a1i 11 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 0 ∈ V) |
22 | id 22 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵) | |
23 | 14, 19, 21, 22 | fsuppco 9440 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡(𝑥 ∈ (ℕ0 ↑m 1o) ↦ (𝑥‘∅))) finSupp 0 ) |
24 | 9, 23 | eqbrtrd 5170 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5688 ∘ ccom 5693 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1oc1o 8498 ↑m cmap 8865 finSupp cfsupp 9399 ℕ0cn0 12524 Basecbs 17245 0gc0g 17486 mPoly cmpl 21944 Poly1cpl1 22194 coe1cco1 22195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-ple 17318 df-psr 21947 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-ply1 22199 df-coe1 22200 |
This theorem is referenced by: coe1fsupp 22232 mptcoe1fsupp 22233 ply1coefsupp 22317 evls1fpws 22389 mptcoe1matfsupp 22824 mp2pm2mplem4 22831 plypf1 26266 |
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