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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 19791 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
12 | mplcoe2.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
13 | 1 | mplcrng 21222 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing) |
14 | 5, 9, 13 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ CRing) |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑃 ∈ CRing) |
16 | eqid 2740 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
17 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝐼 ∈ 𝑊) |
18 | 11 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑅 ∈ Ring) |
19 | simprr 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑦 ∈ 𝐼) | |
20 | 1, 8, 16, 17, 18, 19 | mvrcl 21217 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑦) ∈ (Base‘𝑃)) |
21 | simprl 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑥 ∈ 𝐼) | |
22 | 1, 8, 16, 17, 18, 21 | mvrcl 21217 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑥) ∈ (Base‘𝑃)) |
23 | eqid 2740 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
24 | 6, 23 | mgpplusg 19720 | . . . . . 6 ⊢ (.r‘𝑃) = (+g‘𝐺) |
25 | 24 | eqcomi 2749 | . . . . 5 ⊢ (+g‘𝐺) = (.r‘𝑃) |
26 | 16, 25 | crngcom 19797 | . . . 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 1370 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
28 | 27 | ralrimivva 3117 | . 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 21237 | 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 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ifcif 4465 ↦ cmpt 5162 ◡ccnv 5588 “ cima 5592 ‘cfv 6431 (class class class)co 7269 ↑m cmap 8596 Fincfn 8714 ℕcn 11971 ℕ0cn0 12231 Basecbs 16908 +gcplusg 16958 .rcmulr 16959 0gc0g 17146 Σg cgsu 17147 .gcmg 18696 mulGrpcmgp 19716 1rcur 19733 Ringcrg 19779 CRingccrg 19780 mVar cmvr 21104 mPoly cmpl 21105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-ofr 7526 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-seq 13718 df-hash 14041 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-tset 16977 df-0g 17148 df-gsum 17149 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mhm 18426 df-submnd 18427 df-grp 18576 df-minusg 18577 df-mulg 18697 df-subg 18748 df-ghm 18828 df-cntz 18919 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-srg 19738 df-ring 19781 df-cring 19782 df-subrg 20018 df-psr 21108 df-mvr 21109 df-mpl 21110 |
This theorem is referenced by: mplbas2 21239 |
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