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Mirrors > Home > MPE Home > Th. List > mplmul | Structured version Visualization version GIF version |
Description: The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplmul.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplmul.b | ⊢ 𝐵 = (Base‘𝑃) |
mplmul.m | ⊢ · = (.r‘𝑅) |
mplmul.t | ⊢ ∙ = (.r‘𝑃) |
mplmul.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mplmul.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mplmul.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
mplmul | ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝐹‘𝑥) · (𝐺‘(𝑘 ∘f − 𝑥))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | eqid 2737 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
3 | mplmul.m | . 2 ⊢ · = (.r‘𝑅) | |
4 | mplmul.t | . . 3 ⊢ ∙ = (.r‘𝑃) | |
5 | mplmul.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
6 | 5 | fvexi 6853 | . . . 4 ⊢ 𝐵 ∈ V |
7 | mplmul.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
8 | 7, 1, 5 | mplval2 21354 | . . . . 5 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s 𝐵) |
9 | eqid 2737 | . . . . 5 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘(𝐼 mPwSer 𝑅)) | |
10 | 8, 9 | ressmulr 17148 | . . . 4 ⊢ (𝐵 ∈ V → (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑃)) |
11 | 6, 10 | ax-mp 5 | . . 3 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑃) |
12 | 4, 11 | eqtr4i 2768 | . 2 ⊢ ∙ = (.r‘(𝐼 mPwSer 𝑅)) |
13 | mplmul.d | . 2 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
14 | 7, 1, 5, 2 | mplbasss 21355 | . . 3 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)) |
15 | mplmul.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
16 | 14, 15 | sselid 3940 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
17 | mplmul.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
18 | 14, 17 | sselid 3940 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
19 | 1, 2, 3, 12, 13, 16, 18 | psrmulfval 21306 | 1 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝐹‘𝑥) · (𝐺‘(𝑘 ∘f − 𝑥))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3405 Vcvv 3443 class class class wbr 5103 ↦ cmpt 5186 ◡ccnv 5630 “ cima 5634 ‘cfv 6493 (class class class)co 7351 ∘f cof 7607 ∘r cofr 7608 ↑m cmap 8723 Fincfn 8841 ≤ cle 11148 − cmin 11343 ℕcn 12111 ℕ0cn0 12371 Basecbs 17043 .rcmulr 17094 Σg cgsu 17282 mPwSer cmps 21259 mPoly cmpl 21261 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-tset 17112 df-psr 21264 df-mpl 21266 |
This theorem is referenced by: mplmonmul 21389 mhpmulcl 21491 mdegmullem 25395 |
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