splysubrg - Mathbox for Thierry Arnoux

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Theorem splysubrg 33959

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 33958	. 2 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
8		eqid 2762	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓))
9	1, 2, 3, 4, 5	mplvrpmga 33944	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
10		coeq2 5843	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑒))
11	10	fveq2d 6885	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑒)))
12	11	mpteq2dv 5204	. . . . . 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 6880	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘(𝑥 ∘ 𝑒)) = (𝑔‘(𝑥 ∘ 𝑒)))
14	13	mpteq2dv 5204	. . . . . 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 7507	. . . . 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 2785	. . . 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 7420	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑑𝐴𝑓) = (𝑑𝐴𝑔))
19	18	cbvmptv 5214	. . . 4 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑔 ∈ 𝑀 ↦ (𝑑𝐴𝑔))
20		eqid 2762	. . . 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 33946	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) ∈ ((𝐼 mPoly 𝑅) RingHom (𝐼 mPoly 𝑅)))
24	2, 3, 8, 9, 23	fxpsubrg 33503	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
25	7, 24	eqeltrd 2862	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 {crab 3415 class class class wbr 5108 ↦ cmpt 5191 ∘ ccom 5664 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8822 finSupp cfsupp 9319 0cc0 11106 ℕ₀cn0 12510 Basecbs 17275 SymGrpcsymg 19445 Ringcrg 20321 SubRingcsubrg 20679 mPoly cmpl 22067 FixPtscfxp 33492 SymPolycsply 33954

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-fzo 13690 df-seq 14045 df-hash 14374 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-hom 17340 df-cco 17341 df-0g 17500 df-gsum 17501 df-prds 17506 df-pws 17508 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-efmnd 18934 df-grp 19009 df-minusg 19010 df-mulg 19140 df-subg 19195 df-ghm 19290 df-ga 19366 df-cntz 19393 df-symg 19446 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-psr 22070 df-mpl 22072 df-fxp 33493 df-sply 33956

This theorem is used by: (None)

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