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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 2726 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | eqid 2726 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
3 | mplmul.m | . 2 ⊢ · = (.r‘𝑅) | |
4 | mplmul.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
5 | mplmul.t | . . 3 ⊢ ∙ = (.r‘𝑃) | |
6 | 4, 1, 5 | mplmulr 22017 | . 2 ⊢ ∙ = (.r‘(𝐼 mPwSer 𝑅)) |
7 | mplmul.d | . 2 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
8 | mplmul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | 4, 1, 8, 2 | mplbasss 22006 | . . 3 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)) |
10 | mplmul.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
11 | 9, 10 | sselid 3977 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
12 | mplmul.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
13 | 9, 12 | sselid 3977 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
14 | 1, 2, 3, 6, 7, 11, 13 | psrmulfval 21952 | 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 1534 ∈ wcel 2099 {crab 3419 class class class wbr 5153 ↦ cmpt 5236 ◡ccnv 5681 “ cima 5685 ‘cfv 6554 (class class class)co 7424 ∘f cof 7688 ∘r cofr 7689 ↑m cmap 8855 Fincfn 8974 ≤ cle 11299 − cmin 11494 ℕcn 12264 ℕ0cn0 12524 Basecbs 17213 .rcmulr 17267 Σg cgsu 17455 mPwSer cmps 21901 mPoly cmpl 21903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 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 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-tset 17285 df-psr 21906 df-mpl 21908 |
This theorem is referenced by: mplmonmul 22043 mhpmulcl 22143 rhmcomulmpl 22373 mdegmullem 26105 |
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