mplgsum - Mathbox for Thierry Arnoux

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Theorem mplgsum 33722

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 21956	. . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
7		ovexd 7396	. . 3 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ V)
8		mplgsum.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
9	3, 4, 5, 1	mplbasss 21957	. . . 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 20182	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
15		mplgsum.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}
16	15	psrbasfsupp 33697	. . . . . 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 21918	. . . . 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 21966	. . . . 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 21977	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
24	12, 14, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp)
25	5, 20	grpidcl 18900	. . . . 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 21917	. . . . . 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 18904	. . . 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 18905	. . . 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 18612	. 2 ⊢ (𝜑 → ((𝐼 mPwSer 𝑅) Σ_g 𝐹) = (𝑃 Σ_g 𝐹))
35	11, 10	fssd 6680	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘(𝐼 mPwSer 𝑅)))
36	4, 1, 13, 12, 15, 8, 35	psrgsum 33717	. 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 3400 Vcvv 3441 ⊆ wss 3902 {csn 4581 class class class wbr 5099 ↦ cmpt 5180 × cxp 5623 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8768 Fincfn 8888 finSupp cfsupp 9269 0cc0 11031 ℕ₀cn0 12406 Basecbs 17141 +_gcplusg 17182 0_gc0g 17364 Σ_g cgsu 17365 Grpcgrp 18868 Ringcrg 20173 mPwSer cmps 21865 mPoly cmpl 21867

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-ofr 7626 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-oi 9420 df-card 9856 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-z 12494 df-dec 12613 df-uz 12757 df-fz 13429 df-fzo 13576 df-seq 13930 df-hash 14259 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-hom 17206 df-cco 17207 df-0g 17366 df-gsum 17367 df-prds 17372 df-pws 17374 df-mre 17510 df-mrc 17511 df-acs 17513 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-submnd 18714 df-grp 18871 df-minusg 18872 df-mulg 19003 df-subg 19058 df-ghm 19147 df-cntz 19251 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-psr 21870 df-mpl 21872

This theorem is referenced by: esplyfval1 33742

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