splysubrg - Mathbox for Thierry Arnoux

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Theorem splysubrg 33920

Description: The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.)

**Hypotheses**
Ref	Expression
splyval.s	⊢ 𝑆 = (SymGrp‘𝐼)
splyval.p	⊢ 𝑃 = (Base‘𝑆)
splyval.m	⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
splyval.a	⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
splyval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
splysubrg.r	⊢ (𝜑 → 𝑅 ∈ Ring)

**Assertion**
Ref	Expression
splysubrg	⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

Distinct variable groups: 𝐴,𝑑,𝑓,𝑥 𝐼,𝑑,𝑓,ℎ,𝑥 𝑀,𝑑,𝑓,𝑥 𝑃,𝑑,𝑓,𝑥 𝑅,𝑑,𝑓,𝑥,ℎ 𝑆,𝑑,𝑓 𝑥,𝑉 𝜑,𝑑,𝑓,𝑥 Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ) 𝑃(ℎ) 𝑆(𝑥,ℎ) 𝑀(ℎ) 𝑉(𝑓,ℎ,𝑑)

Proof of Theorem splysubrg Dummy variables 𝑒 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		splyval.s	. . 3 ⊢ 𝑆 = (SymGrp‘𝐼)
2		splyval.p	. . 3 ⊢ 𝑃 = (Base‘𝑆)
3		splyval.m	. . 3 ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
4		splyval.a	. . 3 ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
5		splyval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		splysubrg.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	1, 2, 3, 4, 5, 6	splyval 33919	. 2 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
8		eqid 2770	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓))
9	1, 2, 3, 4, 5	mplvrpmga 33905	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
10		coeq2 5848	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑒))
11	10	fveq2d 6889	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑒)))
12	11	mpteq2dv 5210	. . . . . 6 ⊢ (𝑑 = 𝑒 → (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑒))))
13		fveq1 6884	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘(𝑥 ∘ 𝑒)) = (𝑔‘(𝑥 ∘ 𝑒)))
14	13	mpteq2dv 5210	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑒))) = (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑒))))
15	12, 14	cbvmpov 7509	. . . . 5 ⊢ (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) = (𝑒 ∈ 𝑃, 𝑔 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑒))))
16	4, 15	eqtri 2793	. . . 4 ⊢ 𝐴 = (𝑒 ∈ 𝑃, 𝑔 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑒))))
17	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝐼 ∈ 𝑉)
18		oveq2 7422	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑑𝐴𝑓) = (𝑑𝐴𝑔))
19	18	cbvmptv 5220	. . . 4 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑔 ∈ 𝑀 ↦ (𝑑𝐴𝑔))
20		eqid 2770	. . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
21	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑅 ∈ Ring)
22		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑑 ∈ 𝑃)
23	1, 2, 3, 16, 17, 19, 20, 21, 22	mplvrpmrhm 33907	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) ∈ ((𝐼 mPoly 𝑅) RingHom (𝐼 mPoly 𝑅)))
24	2, 3, 8, 9, 23	fxpsubrg 33464	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
25	7, 24	eqeltrd 2870	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1568 ∈ wcel 2150 {crab 3423 class class class wbr 5114 ↦ cmpt 5197 ∘ ccom 5669 ‘cfv 6540 (class class class)co 7414 ∈ cmpo 7416 ↑_m cmap 8827 finSupp cfsupp 9324 0cc0 11103 ℕ₀cn0 12507 Basecbs 17272 SymGrpcsymg 19442 Ringcrg 20318 SubRingcsubrg 20657 mPoly cmpl 22039 FixPtscfxp 33453 SymPolycsply 33915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7678 df-ofr 7679 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8899 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fsupp 9325 df-sup 9405 df-oi 9475 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-z 12595 df-dec 12715 df-uz 12866 df-fz 13539 df-fzo 13686 df-seq 14041 df-hash 14370 df-struct 17210 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-hom 17337 df-cco 17338 df-0g 17497 df-gsum 17498 df-prds 17503 df-pws 17505 df-mre 17641 df-mrc 17642 df-acs 17644 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-mhm 18844 df-submnd 18845 df-efmnd 18931 df-grp 19006 df-minusg 19007 df-mulg 19137 df-subg 19192 df-ghm 19287 df-ga 19363 df-cntz 19390 df-symg 19443 df-cmn 19855 df-abl 19856 df-mgp 20220 df-rng 20234 df-ur 20267 df-ring 20320 df-rhm 20557 df-subrng 20634 df-subrg 20658 df-psr 22042 df-mpl 22044 df-fxp 33454 df-sply 33917

This theorem is referenced by: (None)

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