mplgsum - Mathbox for Thierry Arnoux

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Theorem mplgsum 33694

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 2737	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
2		eqid 2737	. . 3 ⊢ (+_g‘(𝐼 mPwSer 𝑅)) = (+_g‘(𝐼 mPwSer 𝑅))
3		mplgsum.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4		eqid 2737	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5		mplgsum.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
6	3, 4, 5	mplval2 21971	. . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
7		ovexd 7399	. . 3 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ V)
8		mplgsum.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
9	3, 4, 5, 1	mplbasss 21972	. . . 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 20220	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
15		mplgsum.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
16	15	psrbasfsupp 33669	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
17		eqid 2737	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18		eqid 2737	. . . . . 6 ⊢ (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘(𝐼 mPwSer 𝑅))
19	4, 12, 14, 16, 17, 18	psr0 21933	. . . . 5 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (𝐷 × {(0_g‘𝑅)}))
20		eqid 2737	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
21	3, 16, 17, 20, 12, 14	mpl0 21981	. . . . 5 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × {(0_g‘𝑅)}))
22	19, 21	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) = (0_g‘𝑃))
23	3	mplgrp 21992	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
24	12, 14, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp)
25	5, 20	grpidcl 18938	. . . . 5 ⊢ (𝑃 ∈ Grp → (0_g‘𝑃) ∈ 𝐵)
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → (0_g‘𝑃) ∈ 𝐵)
27	22, 26	eqeltrd 2837	. . 3 ⊢ (𝜑 → (0_g‘(𝐼 mPwSer 𝑅)) ∈ 𝐵)
28	4, 12, 14	psrgrp 21932	. . . . . 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 18942	. . . 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 18943	. . . 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 18647	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑃 Σ_g 𝐹))
35	11, 10	fssd 6683	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐼 mPwSer 𝑅)))
36	4, 1, 13, 12, 15, 8, 35	psrgsum 33689	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
37	34, 36	eqtr3d 2774	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 3390 Vcvv 3430 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 × cxp 5626 ⟶wf 6492 ‘cfv 6496 (class class class)co 7364 ↑_m cmap 8770 Fincfn 8890 finSupp cfsupp 9271 0cc0 11035 ℕ₀cn0 12434 Basecbs 17176 +_gcplusg 17217 0_gc0g 17399 Σ_g cgsu 17400 Grpcgrp 18906 Ringcrg 20211 mPwSer cmps 21881 mPoly cmpl 21883

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 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-se 5582 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-of 7628 df-ofr 7629 df-om 7815 df-1st 7939 df-2nd 7940 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 9860 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-7 12246 df-8 12247 df-9 12248 df-n0 12435 df-z 12522 df-dec 12642 df-uz 12786 df-fz 13459 df-fzo 13606 df-seq 13961 df-hash 14290 df-struct 17114 df-sets 17131 df-slot 17149 df-ndx 17161 df-base 17177 df-ress 17198 df-plusg 17230 df-mulr 17231 df-sca 17233 df-vsca 17234 df-ip 17235 df-tset 17236 df-ple 17237 df-ds 17239 df-hom 17241 df-cco 17242 df-0g 17401 df-gsum 17402 df-prds 17407 df-pws 17409 df-mre 17545 df-mrc 17546 df-acs 17548 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18748 df-submnd 18749 df-grp 18909 df-minusg 18910 df-mulg 19041 df-subg 19096 df-ghm 19185 df-cntz 19289 df-cmn 19754 df-abl 19755 df-mgp 20119 df-rng 20131 df-ur 20160 df-ring 20213 df-psr 21886 df-mpl 21888

This theorem is referenced by: esplyfval1 33714

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