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Theorem mplcoe2 22024

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.)

**Hypotheses**
Ref	Expression
mplcoe1.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplcoe1.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
mplcoe1.z	⊢ 0 = (0_g‘𝑅)
mplcoe1.o	⊢ 1 = (1_r‘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)

**Assertion**
Ref	Expression
mplcoe2	⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))

Distinct variable groups: ↑ ,𝑘,𝑦 1 ,𝑘,𝑦 𝑘,𝐺 𝑓,𝑘,𝑦,𝐼 𝜑,𝑘,𝑦 𝑅,𝑓,𝑦 𝐷,𝑘,𝑦 𝑃,𝑘 𝑘,𝑉 0 ,𝑓,𝑘,𝑦 𝑓,𝑌,𝑘,𝑦 𝑘,𝑊,𝑦 𝑦,𝐺 𝑦,𝑉 𝑦, ↑ Allowed substitution hints: 𝜑(𝑓) 𝐷(𝑓) 𝑃(𝑦,𝑓) 𝑅(𝑘) 1 (𝑓) ↑ (𝑓) 𝐺(𝑓) 𝑉(𝑓) 𝑊(𝑓)

Proof of Theorem mplcoe2 Dummy variable 𝑥 is distinct from all other variables.

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 = (0_g‘𝑅)
4		mplcoe1.o	. 2 ⊢ 1 = (1_r‘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 20224	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		mplcoe2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
13	1	mplcrng 22002	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
14	5, 9, 13	syl2anc 590	. . . . 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 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑦 ∈ 𝐼)
20	1, 8, 16, 17, 18, 19	mvrcl 21973	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑦) ∈ (Base‘𝑃))
21		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑥 ∈ 𝐼)
22	1, 8, 16, 17, 18, 21	mvrcl 21973	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑥) ∈ (Base‘𝑃))
23		eqid 2740	. . . . . . 7 ⊢ (._r‘𝑃) = (._r‘𝑃)
24	6, 23	mgpplusg 20123	. . . . . 6 ⊢ (._r‘𝑃) = (+_g‘𝐺)
25	24	eqcomi 2749	. . . . 5 ⊢ (+_g‘𝐺) = (._r‘𝑃)
26	16, 25	crngcom 20230	. . . 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 1379	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → ((𝑉‘𝑦)(+_g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+_g‘𝐺)(𝑉‘𝑦)))
28	27	ralrimivva 3183	. 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 22023	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3392 ifcif 4461 ↦ cmpt 5160 ^◡ccnv 5624 “ cima 5628 ‘cfv 6492 (class class class)co 7363 ↑_m cmap 8770 Fincfn 8890 ℕcn 12172 ℕ₀cn0 12435 Basecbs 17177 +_gcplusg 17218 ._rcmulr 17219 0_gc0g 17400 Σ_g cgsu 17401 ._gcmg 19041 mulGrpcmgp 20119 1_rcur 20160 Ringcrg 20212 CRingccrg 20213 mVar cmvr 21887 mPoly cmpl 21888

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-ofr 7628 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-fzo 13607 df-seq 13962 df-hash 14291 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17402 df-gsum 17403 df-prds 17408 df-pws 17410 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-mulg 19042 df-subg 19097 df-ghm 19186 df-cntz 19290 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-srg 20166 df-ring 20214 df-cring 20215 df-subrng 20525 df-subrg 20549 df-psr 21891 df-mvr 21892 df-mpl 21893

This theorem is referenced by: mplbas2 22025 selvvvval 22125

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