Theorem splyval 33582

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 33580	. . 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 7355	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
4	3	fveq2d 6826	. . . . 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 2784	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝑀)
7		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (SymGrp‘𝑖) = (SymGrp‘𝐼))
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑖) = (SymGrp‘𝐼))
9		splyval.s	. . . . . . . . 9 ⊢ 𝑆 = (SymGrp‘𝐼)
10	8, 9	eqtr4di 2784	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (SymGrp‘𝑖) = 𝑆)
11	10	fveq2d 6826	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑖)) = (Base‘𝑆))
12		splyval.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝑆)
13	11, 12	eqtr4di 2784	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(SymGrp‘𝑖)) = 𝑃)
14		oveq2 7354	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
16	15	rabeqdv 3410	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
17	16	mpteq1d 5179	. . . . . 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 7421	. . . . 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 2784	. . . 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 7364	. . 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 481	. 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 3460	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
25		splyval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
26	25	elexd 3460	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
27		ovexd 7381	. 2 ⊢ (𝜑 → (𝑀FixPts𝐴) ∈ V)
28	2, 22, 24, 26, 27	ovmpod 7498	1 ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   class class class wbr 5089   ↦ cmpt 5170   ∘ ccom 5618  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   ↑_m cmap 8750   finSupp cfsupp 9245  0cc0 11006  ℕ₀cn0 12381  Basecbs 17120  SymGrpcsymg 19281   mPoly cmpl 21843  FixPtscfxp 33132  SymPolycsply 33578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-sply 33580

This theorem is referenced by:  splysubrg  33583  issply  33584