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Theorem mplgsum 33747

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 2741	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
2		eqid 2741	. . 3 ⊢ (+_g‘(𝐼 mPwSer 𝑅)) = (+_g‘(𝐼 mPwSer 𝑅))
3		mplgsum.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4		eqid 2741	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5		mplgsum.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
6	3, 4, 5	mplval2 21973	. . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
7		ovexd 7394	. . 3 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ V)
8		mplgsum.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
9	3, 4, 5, 1	mplbasss 21974	. . . 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 20217	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
15		mplgsum.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
16	15	psrbasfsupp 33705	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
17		eqid 2741	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18		eqid 2741	. . . . . 6 ⊢ (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘(𝐼 mPwSer 𝑅))
19	4, 12, 14, 16, 17, 18	psr0 21935	. . . . 5 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (𝐷 × {(0_g‘𝑅)}))
20		eqid 2741	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
21	3, 16, 17, 20, 12, 14	mpl0 21983	. . . . 5 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × {(0_g‘𝑅)}))
22	19, 21	eqtr4d 2779	. . . 4 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘𝑃))
23	3	mplgrp 21994	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
24	12, 14, 23	syl2anc 591	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp)
25	5, 20	grpidcl 18936	. . . . 5 ⊢ (𝑃 ∈ Grp → (0_g‘𝑃) ∈ 𝐵)
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → (0_g‘𝑃) ∈ 𝐵)
27	22, 26	eqeltrd 2841	. . 3 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) ∈ 𝐵)
28	4, 12, 14	psrgrp 21934	. . . . . 6 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ Grp)
29	28	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝐼 mPwSer 𝑅) ∈ Grp)
30		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅)))
31	1, 2, 18, 29, 30	grplidd 18940	. . . 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 18941	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑥(+_g‘(𝐼 mPwSer 𝑅))(0_g‘(𝐼 mPwSer 𝑅))) = 𝑥)
33	31, 32	jca 517	. . 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 18645	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑃 Σ_g 𝐹))
35	11, 10	fssd 6675	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐼 mPwSer 𝑅)))
36	4, 1, 13, 12, 15, 8, 35	psrgsum 33742	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
37	34, 36	eqtr3d 2778	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 397 = wceq 1548 ∈ wcel 2121 {crab 3393 Vcvv 3433 ⊆ wss 3884 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 × cxp 5618 ⟶wf 6484 ‘cfv 6488 (class class class)co 7359 ↑_m cmap 8767 Fincfn 8887 finSupp cfsupp 9268 0cc0 11034 ℕ₀cn0 12432 Basecbs 17174 +_gcplusg 17215 0_gc0g 17397 Σ_g cgsu 17398 Grpcgrp 18904 Ringcrg 20208 mPwSer cmps 21882 mPoly cmpl 21884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 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 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-ofr 7624 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-fzo 13604 df-seq 13959 df-hash 14288 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-psr 21887 df-mpl 21889

This theorem is referenced by: esplyfval1 33767

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