Theorem splyval 33958

Description: The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-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)
splyval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
splyval	⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))

Distinct variable groups:   𝐼,𝑑,𝑓,ℎ,𝑥   𝑅,𝑑,𝑓

Allowed substitution hints:   𝜑(𝑥, 𝑓, ℎ, 𝑑)   𝐴(𝑥, 𝑓, ℎ, 𝑑)   𝑃(𝑥, 𝑓, ℎ, 𝑑)   𝑅(𝑥, ℎ)   𝑆(𝑥, 𝑓, ℎ, 𝑑)   𝑀(𝑥, 𝑓, ℎ, 𝑑)   𝑉(𝑥, 𝑓, ℎ, 𝑑)   𝑊(𝑥, 𝑓, ℎ, 𝑑)

Proof of Theorem splyval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sply 33956	. . 3 ⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
2	1	a1i 11	. 2 ⊢ (𝜑 → SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))))
3		oveq12 7419	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
4	3	fveq2d 6885	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘(𝐼 mPoly 𝑅)))
5		splyval.m	. . . . 5 ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
6	4, 5	eqtr4di 2816	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝑀)
7		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (SymGrp‘𝑖) = (SymGrp‘𝐼))
8	7	adantr 485	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑖) = (SymGrp‘𝐼))
9		splyval.s	. . . . . . . . 9 ⊢ 𝑆 = (SymGrp‘𝐼)
10	8, 9	eqtr4di 2816	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑖) = 𝑆)
11	10	fveq2d 6885	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑖)) = (Base‘𝑆))
12		splyval.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝑆)
13	11, 12	eqtr4di 2816	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑖)) = 𝑃)
14		oveq2 7418	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
15	14	adantr 485	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
16	15	rabeqdv 3431	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
17	16	mpteq1d 5201	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
18	13, 6, 17	mpoeq123dv 7485	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
19		splyval.a	. . . . 5 ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
20	18, 19	eqtr4di 2816	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) = 𝐴)
21	6, 20	oveq12d 7428	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))) = (𝑀FixPts𝐴))
22	21	adantl 486	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))) = (𝑀FixPts𝐴))
23		splyval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
24	23	elexd 3478	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
25		splyval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
26	25	elexd 3478	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
27		ovexd 7445	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ V)
28	2, 22, 24, 26, 27	ovmpod 7562	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   class class class wbr 5109   ↦ cmpt 5192   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820   finSupp cfsupp 9317  0cc0 11104  ℕ₀cn0 12508  Basecbs 17273  SymGrpcsymg 19443   mPoly cmpl 22065  FixPtscfxp 33492  SymPolycsply 33954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-sply 33956

This theorem is used by:  splysubrg  33959  issply  33960