mplgsum - Mathbox for Thierry Arnoux

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Theorem mplgsum 33685

Description: Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)

**Hypotheses**
Ref	Expression
mplgsum.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplgsum.b	⊢ 𝐵 = (Base‘𝑃)
mplgsum.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplgsum.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mplgsum.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
mplgsum.a	⊢ (𝜑 → 𝐴 ∈ Fin)
mplgsum.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

**Assertion**
Ref	Expression
mplgsum	⊢ (𝜑 → (𝑃 Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))

Distinct variable groups: 𝐴,𝑘,𝑦 𝐷,𝑘,𝑦 𝑘,𝐹,𝑦 ℎ,𝐼 𝑘,𝐼,𝑦 𝑅,𝑘,𝑦 𝜑,𝑘,𝑦 Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ) 𝐵(𝑦,ℎ,𝑘) 𝐷(ℎ) 𝑃(𝑦,ℎ,𝑘) 𝑅(ℎ) 𝐹(ℎ) 𝑉(𝑦,ℎ,𝑘)

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

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
2		eqid 2735	. . 3 ⊢ (+_g‘(𝐼 mPwSer 𝑅)) = (+_g‘(𝐼 mPwSer 𝑅))
3		mplgsum.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4		eqid 2735	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5		mplgsum.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
6	3, 4, 5	mplval2 21963	. . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
7		ovexd 7391	. . 3 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ V)
8		mplgsum.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
9	3, 4, 5, 1	mplbasss 21964	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
11		mplgsum.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12		mplgsum.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
13		mplgsum.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	13	ringgrpd 20212	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
15		mplgsum.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
16	15	psrbasfsupp 33660	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
17		eqid 2735	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18		eqid 2735	. . . . . 6 ⊢ (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘(𝐼 mPwSer 𝑅))
19	4, 12, 14, 16, 17, 18	psr0 21925	. . . . 5 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (𝐷 × {(0_g‘𝑅)}))
20		eqid 2735	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
21	3, 16, 17, 20, 12, 14	mpl0 21973	. . . . 5 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × {(0_g‘𝑅)}))
22	19, 21	eqtr4d 2773	. . . 4 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘𝑃))
23	3	mplgrp 21984	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
24	12, 14, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp)
25	5, 20	grpidcl 18930	. . . . 5 ⊢ (𝑃 ∈ Grp → (0_g‘𝑃) ∈ 𝐵)
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → (0_g‘𝑃) ∈ 𝐵)
27	22, 26	eqeltrd 2835	. . 3 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) ∈ 𝐵)
28	4, 12, 14	psrgrp 21924	. . . . . 6 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ Grp)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝐼 mPwSer 𝑅) ∈ Grp)
30		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅)))
31	1, 2, 18, 29, 30	grplidd 18934	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → ((0_g‘(𝐼 mPwSer 𝑅))(+_g‘(𝐼 mPwSer 𝑅))𝑥) = 𝑥)
32	1, 2, 18, 29, 30	grpridd 18935	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑥(+_g‘(𝐼 mPwSer 𝑅))(0_g‘(𝐼 mPwSer 𝑅))) = 𝑥)
33	31, 32	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (((0_g‘(𝐼 mPwSer 𝑅))(+_g‘(𝐼 mPwSer 𝑅))𝑥) = 𝑥 ∧ (𝑥(+_g‘(𝐼 mPwSer 𝑅))(0_g‘(𝐼 mPwSer 𝑅))) = 𝑥))
34	1, 2, 6, 7, 8, 10, 11, 27, 33	gsumress 18639	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑃 Σ_g 𝐹))
35	11, 10	fssd 6674	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐼 mPwSer 𝑅)))
36	4, 1, 13, 12, 15, 8, 35	psrgsum 33680	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
37	34, 36	eqtr3d 2772	1 ⊢ (𝜑 → (𝑃 Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3387 Vcvv 3427 ⊆ wss 3885 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 × cxp 5618 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ↑_m cmap 8762 Fincfn 8882 finSupp cfsupp 9263 0cc0 11027 ℕ₀cn0 12426 Basecbs 17168 +_gcplusg 17209 0_gc0g 17391 Σ_g cgsu 17392 Grpcgrp 18898 Ringcrg 20203 mPwSer cmps 21873 mPoly cmpl 21875

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-sup 9344 df-oi 9414 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-mulg 19033 df-subg 19088 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-psr 21878 df-mpl 21880

This theorem is referenced by: esplyfval1 33705

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