mplcoe2 - Metamath Proof Explorer

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Theorem mplcoe2 21986

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 20192	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		mplcoe2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
13	1	mplcrng 21970	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
14	5, 9, 13	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ CRing)
15	14	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑃 ∈ CRing)
16		eqid 2728	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
17	5	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝐼 ∈ 𝑊)
18	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑅 ∈ Ring)
19		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑦 ∈ 𝐼)
20	1, 8, 16, 17, 18, 19	mvrcl 21941	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑦) ∈ (Base‘𝑃))
21		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → 𝑥 ∈ 𝐼)
22	1, 8, 16, 17, 18, 21	mvrcl 21941	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑉‘𝑥) ∈ (Base‘𝑃))
23		eqid 2728	. . . . . . 7 ⊢ (._r‘𝑃) = (._r‘𝑃)
24	6, 23	mgpplusg 20085	. . . . . 6 ⊢ (._r‘𝑃) = (+_g‘𝐺)
25	24	eqcomi 2737	. . . . 5 ⊢ (+_g‘𝐺) = (._r‘𝑃)
26	16, 25	crngcom 20198	. . . 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 1368	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → ((𝑉‘𝑦)(+_g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+_g‘𝐺)(𝑉‘𝑦)))
28	27	ralrimivva 3198	. 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 21985	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 394 = wceq 1533 ∈ wcel 2098 {crab 3430 ifcif 4532 ↦ cmpt 5235 ^◡ccnv 5681 “ cima 5685 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8851 Fincfn 8970 ℕcn 12250 ℕ₀cn0 12510 Basecbs 17187 +_gcplusg 17240 ._rcmulr 17241 0_gc0g 17428 Σ_g cgsu 17429 ._gcmg 19030 mulGrpcmgp 20081 1_rcur 20128 Ringcrg 20180 CRingccrg 20181 mVar cmvr 21845 mPoly cmpl 21846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-psr 21849 df-mvr 21850 df-mpl 21851

This theorem is referenced by: mplbas2 21987 selvvvval 41849

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