mplgsum - Mathbox for Thierry Arnoux

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Theorem mplgsum 33702

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 21952	. . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
7		ovexd 7393	. . 3 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ V)
8		mplgsum.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
9	3, 4, 5, 1	mplbasss 21953	. . . 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 20181	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
15		mplgsum.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
16	15	psrbasfsupp 33677	. . . . . 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 21914	. . . . 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 21962	. . . . 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 21973	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
24	12, 14, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp)
25	5, 20	grpidcl 18899	. . . . 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 21913	. . . . . 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 18903	. . . 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 18904	. . . 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 18608	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑃 Σ_g 𝐹))
35	11, 10	fssd 6677	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐼 mPwSer 𝑅)))
36	4, 1, 13, 12, 15, 8, 35	psrgsum 33697	. 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 5620 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ↑_m cmap 8764 Fincfn 8884 finSupp cfsupp 9265 0cc0 11027 ℕ₀cn0 12402 Basecbs 17137 +_gcplusg 17178 0_gc0g 17360 Σ_g cgsu 17361 Grpcgrp 18867 Ringcrg 20172 mPwSer cmps 21861 mPoly cmpl 21863

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 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 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 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-fz 13425 df-fzo 13572 df-seq 13926 df-hash 14255 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-hom 17202 df-cco 17203 df-0g 17362 df-gsum 17363 df-prds 17368 df-pws 17370 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-grp 18870 df-minusg 18871 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-psr 21866 df-mpl 21868

This theorem is referenced by: esplyfval1 33722

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