splysubrg - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > splysubrg		Structured version Visualization version GIF version

Theorem splysubrg 34111

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 34110	. 2 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
8		eqid 2760	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓))
9	1, 2, 3, 4, 5	mplvrpmga 34096	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
10		coeq2 5833	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑒))
11	10	fveq2d 6878	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑒)))
12	11	mpteq2dv 5199	. . . . . 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 6873	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘(𝑥 ∘ 𝑒)) = (𝑔‘(𝑥 ∘ 𝑒)))
14	13	mpteq2dv 5199	. . . . . 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 7504	. . . . 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 2783	. . . 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 7417	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑑𝐴𝑓) = (𝑑𝐴𝑔))
19	18	cbvmptv 5209	. . . 4 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑔 ∈ 𝑀 ↦ (𝑑𝐴𝑔))
20		eqid 2760	. . . 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 34098	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) ∈ ((𝐼 mPoly 𝑅) RingHom (𝐼 mPoly 𝑅)))
24	2, 3, 8, 9, 23	fxpsubrg 33654	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
25	7, 24	eqeltrd 2860	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 2145 {crab 3412 class class class wbr 5103 ↦ cmpt 5186 ∘ ccom 5652 ‘cfv 6528 (class class class)co 7409 ∈ cmpo 7411 ↑_m cmap 8826 finSupp cfsupp 9331 0cc0 11157 ℕ₀cn0 12561 Basecbs 17334 SymGrpcsymg 19530 Ringcrg 20406 SubRingcsubrg 20768 mPoly cmpl 22161 FixPtscfxp 33643 SymPolycsply 34106

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7677 df-ofr 7678 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-3 12361 df-4 12362 df-5 12363 df-6 12364 df-7 12365 df-8 12366 df-9 12367 df-n0 12562 df-z 12649 df-dec 12770 df-uz 12921 df-fz 13595 df-fzo 13743 df-seq 14099 df-hash 14428 df-struct 17272 df-sets 17289 df-slot 17307 df-ndx 17319 df-base 17335 df-ress 17356 df-plusg 17388 df-mulr 17389 df-sca 17391 df-vsca 17392 df-ip 17393 df-tset 17394 df-ple 17395 df-ds 17397 df-hom 17399 df-cco 17400 df-0g 17559 df-gsum 17560 df-prds 17565 df-pws 17567 df-mre 17703 df-mrc 17704 df-acs 17706 df-mgm 18763 df-sgrp 18855 df-mnd 18871 df-mhm 18925 df-submnd 18926 df-efmnd 19012 df-grp 19094 df-minusg 19095 df-mulg 19225 df-subg 19280 df-ghm 19375 df-ga 19451 df-cntz 19478 df-symg 19531 df-cmn 19943 df-abl 19944 df-mgp 20308 df-rng 20322 df-ur 20355 df-ring 20408 df-rhm 20649 df-subrng 20745 df-subrg 20769 df-psr 22164 df-mpl 22166 df-fxp 33644 df-sply 34108

This theorem is used by: (None)

W3C validator