Theorem esplyval 34264

Description: The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)

Hypotheses

Ref	Expression
esplyval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
esplyval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
esplyval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
esplyval	⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))

Distinct variable groups:   𝐼,𝑐,ℎ,𝑘   𝑅,𝑘

Allowed substitution hints:   𝜑(ℎ, 𝑘, 𝑐)   𝐷(ℎ, 𝑘, 𝑐)   𝑅(ℎ, 𝑐)   𝑉(ℎ, 𝑘, 𝑐)   𝑊(ℎ, 𝑘, 𝑐)

Proof of Theorem esplyval

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

Step	Hyp	Ref	Expression
1		df-esply 34260	. . 3 ⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
2	1	a1i 11	. 2 ⊢ (𝜑 → eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘}))))))
3		fveq2 6908	. . . . . 6 ⊢ (𝑟 = 𝑅 → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
4	3	adantl 487	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
5		oveq2 7451	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
6	5	rabeqdv 3431	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
7		esplyval.d	. . . . . . . . 9 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
8	6, 7	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
9	8	fveq2d 6912	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0}) = (𝟭‘𝐷))
10	9	adantr 486	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0}) = (𝟭‘𝐷))
11		fveq2 6908	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝟭‘𝑖) = (𝟭‘𝐼))
12	11	adantr 486	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝟭‘𝑖) = (𝟭‘𝐼))
13		pweq 4574	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → 𝒫 𝑖 = 𝒫 𝐼)
14	13	rabeqdv 3431	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘} = {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})
15	14	adantr 486	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘} = {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})
16	12, 15	imaeq12d 6065	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘}) = ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘}))
17	10, 16	fveq12d 6915	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})) = ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))
18	4, 17	coeq12d 5848	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘}))) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘}))))
19	18	mpteq2dv 5202	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))
20	19	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))
21		esplyval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
22	21	elexd 3477	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23		esplyval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
24	23	elexd 3477	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
25		nn0ex 12639	. . . 4 ⊢ ℕ0 ∈ V
26	25	mptex 7252	. . 3 ⊢ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))) ∈ V
27	26	a1i 11	. 2 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))) ∈ V)
28	2, 20, 22, 24, 27	ovmpod 7596	1 ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189   “ cima 5658   ∘ ccom 5659  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   ↑_m cmap 8875   finSupp cfsupp 9381  0cc0 11227  𝟭cind 12347  ℕ₀cn0 12633  ♯chash 14501  ℤRHomczrh 21835  eSymPolycesply 34258

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-pr 5395  ax-un 7775  ax-cnex 11283  ax-1cn 11285  ax-addcl 11287

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-ral 3081  df-rex 3091  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-op 4839  df-uni 4871  df-iun 4956  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-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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-nn 12363  df-n0 12634  df-esply 34260

This theorem is used by:  esplyfval  34265  esplyfval0  34266