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Mirrors > Home > MPE Home > Th. List > mplcoe2 | Structured version Visualization version GIF version |
Description: Decompose a monomial into a finite product of powers of variables. (The assumption that 𝑅 is a commutative ring is not strictly necessary, because the submonoid of monomials is in the center of the multiplicative monoid of polynomials, but it simplifies the proof.) (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2019.) |
Ref | Expression |
---|---|
mplcoe1.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplcoe1.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplcoe1.z | ⊢ 0 = (0g‘𝑅) |
mplcoe1.o | ⊢ 1 = (1r‘𝑅) |
mplcoe1.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mplcoe2.g | ⊢ 𝐺 = (mulGrp‘𝑃) |
mplcoe2.m | ⊢ ↑ = (.g‘𝐺) |
mplcoe2.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mplcoe2.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
mplcoe2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
mplcoe2 | ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplcoe1.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplcoe1.d | . 2 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
3 | mplcoe1.z | . 2 ⊢ 0 = (0g‘𝑅) | |
4 | mplcoe1.o | . 2 ⊢ 1 = (1r‘𝑅) | |
5 | mplcoe1.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | mplcoe2.g | . 2 ⊢ 𝐺 = (mulGrp‘𝑃) | |
7 | mplcoe2.m | . 2 ⊢ ↑ = (.g‘𝐺) | |
8 | mplcoe2.v | . 2 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
9 | mplcoe2.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | crngring 18765 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
12 | mplcoe2.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
13 | 1 | mplcrng 19667 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing) |
14 | 5, 9, 13 | syl2anc 565 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ CRing) |
15 | 14 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑃 ∈ CRing) |
16 | eqid 2771 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
17 | 5 | adantr 466 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝐼 ∈ 𝑊) |
18 | 11 | adantr 466 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑅 ∈ Ring) |
19 | simprr 748 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑦 ∈ 𝐼) | |
20 | 1, 8, 16, 17, 18, 19 | mvrcl 19663 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑦) ∈ (Base‘𝑃)) |
21 | simprl 746 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑥 ∈ 𝐼) | |
22 | 1, 8, 16, 17, 18, 21 | mvrcl 19663 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑥) ∈ (Base‘𝑃)) |
23 | eqid 2771 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
24 | 6, 23 | mgpplusg 18700 | . . . . . 6 ⊢ (.r‘𝑃) = (+g‘𝐺) |
25 | 24 | eqcomi 2780 | . . . . 5 ⊢ (+g‘𝐺) = (.r‘𝑃) |
26 | 16, 25 | crngcom 18769 | . . . 4 ⊢ ((𝑃 ∈ CRing ∧ (𝑉‘𝑦) ∈ (Base‘𝑃) ∧ (𝑉‘𝑥) ∈ (Base‘𝑃)) → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
27 | 15, 20, 22, 26 | syl3anc 1476 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
28 | 27 | ralrimivva 3120 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
29 | 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 28 | mplcoe5 19682 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {crab 3065 ifcif 4225 ↦ cmpt 4863 ◡ccnv 5248 “ cima 5252 ‘cfv 6031 (class class class)co 6792 ↑𝑚 cmap 8008 Fincfn 8108 ℕcn 11221 ℕ0cn0 11493 Basecbs 16063 +gcplusg 16148 .rcmulr 16149 0gc0g 16307 Σg cgsu 16308 .gcmg 17747 mulGrpcmgp 18696 1rcur 18708 Ringcrg 18754 CRingccrg 18755 mVar cmvr 19566 mPoly cmpl 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-ofr 7044 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-ixp 8062 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-oi 8570 df-card 8964 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-fzo 12673 df-seq 13008 df-hash 13321 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-tset 16167 df-0g 16309 df-gsum 16310 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-submnd 17543 df-grp 17632 df-minusg 17633 df-mulg 17748 df-subg 17798 df-ghm 17865 df-cntz 17956 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-srg 18713 df-ring 18756 df-cring 18757 df-subrg 18987 df-psr 19570 df-mvr 19571 df-mpl 19572 |
This theorem is referenced by: mplbas2 19684 |
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