splysubrg - Mathbox for Thierry Arnoux

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Theorem splysubrg 33697

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 33696	. 2 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
8		eqid 2735	. . 3 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓))
9	1, 2, 3, 4, 5	mplvrpmga 33689	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
10		coeq2 5806	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑒))
11	10	fveq2d 6837	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑒)))
12	11	mpteq2dv 5191	. . . . . 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 6832	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘(𝑥 ∘ 𝑒)) = (𝑔‘(𝑥 ∘ 𝑒)))
14	13	mpteq2dv 5191	. . . . . 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 7453	. . . . 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 2758	. . . 4 ⊢ 𝐴 = (𝑒 ∈ 𝑃, 𝑔 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑒))))
17	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝐼 ∈ 𝑉)
18		oveq2 7366	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑑𝐴𝑓) = (𝑑𝐴𝑔))
19	18	cbvmptv 5201	. . . 4 ⊢ (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) = (𝑔 ∈ 𝑀 ↦ (𝑑𝐴𝑔))
20		eqid 2735	. . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
21	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑅 ∈ Ring)
22		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → 𝑑 ∈ 𝑃)
23	1, 2, 3, 16, 17, 19, 20, 21, 22	mplvrpmrhm 33691	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑃) → (𝑓 ∈ 𝑀 ↦ (𝑑𝐴𝑓)) ∈ ((𝐼 mPoly 𝑅) RingHom (𝐼 mPoly 𝑅)))
24	2, 3, 8, 9, 23	fxpsubrg 33235	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
25	7, 24	eqeltrd 2835	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3398 class class class wbr 5097 ↦ cmpt 5178 ∘ ccom 5627 ‘cfv 6491 (class class class)co 7358 ∈ cmpo 7360 ↑_m cmap 8765 finSupp cfsupp 9266 0cc0 11028 ℕ₀cn0 12403 Basecbs 17138 SymGrpcsymg 19300 Ringcrg 20170 SubRingcsubrg 20504 mPoly cmpl 21864 FixPtscfxp 33224 SymPolycsply 33692

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-sup 9347 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-fz 13426 df-fzo 13573 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-efmnd 18796 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 df-ghm 19144 df-ga 19221 df-cntz 19248 df-symg 19301 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-psr 21867 df-mpl 21869 df-fxp 33225 df-sply 33694

This theorem is referenced by: (None)

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