splysubrg - Mathbox for Thierry Arnoux

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Theorem splysubrg 34259

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 34258	. 2 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
8		eqid 2763	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓))
9	1, 2, 3, 4, 5	mplvrpmga 34244	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
10		coeq2 5841	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑒))
11	10	fveq2d 6909	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑒)))
12	11	mpteq2dv 5202	. . . . . 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 6904	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘(𝑥 ∘ 𝑒)) = (𝑔‘(𝑥 ∘ 𝑒)))
14	13	mpteq2dv 5202	. . . . . 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 7535	. . . . 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 2786	. . . 4 ⊢ 𝐴 = (𝑒 ∈ 𝑃, 𝑔 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑒))))
17	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝐼 ∈ 𝑉)
18		oveq2 7448	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑑𝐴𝑓) = (𝑑𝐴𝑔))
19	18	cbvmptv 5212	. . . 4 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑔 ∈ 𝑀 ↦ (𝑑𝐴𝑔))
20		eqid 2763	. . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
21	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑅 ∈ Ring)
22		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑑 ∈ 𝑃)
23	1, 2, 3, 16, 17, 19, 20, 21, 22	mplvrpmrhm 34246	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) ∈ ((𝐼 mPoly 𝑅) RingHom (𝐼 mPoly 𝑅)))
24	2, 3, 8, 9, 23	fxpsubrg 33802	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
25	7, 24	eqeltrd 2864	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 {crab 3416 class class class wbr 5106 ↦ cmpt 5189 ∘ ccom 5659 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 ↑_m cmap 8872 finSupp cfsupp 9378 0cc0 11224 ℕ₀cn0 12630 Basecbs 17413 SymGrpcsymg 19610 Ringcrg 20486 SubRingcsubrg 20848 mPoly cmpl 22241 FixPtscfxp 33791 SymPolycsply 34254

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-ofr 7715 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-fz 13664 df-fzo 13813 df-seq 14169 df-hash 14499 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-submnd 19005 df-efmnd 19091 df-grp 19173 df-minusg 19174 df-mulg 19304 df-subg 19359 df-ghm 19454 df-ga 19531 df-cntz 19558 df-symg 19611 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-rhm 20729 df-subrng 20825 df-subrg 20849 df-psr 22244 df-mpl 22246 df-fxp 33792 df-sply 34256

This theorem is used by: (None)

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