Theorem esplyval 33731

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 33727	. . 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 6835	. . . . . 6 ⊢ (𝑟 = 𝑅 → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
4	3	adantl 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℤRHom‘𝑟) = (ℤRHom‘𝑅))
5		oveq2 7369	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
6	5	rabeqdv 3415	. . . . . . . . 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 2790	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} = 𝐷)
9	8	fveq2d 6839	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0}) = (𝟭‘𝐷))
10	9	adantr 480	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0}) = (𝟭‘𝐷))
11		fveq2 6835	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝟭‘𝑖) = (𝟭‘𝐼))
12	11	adantr 480	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝟭‘𝑖) = (𝟭‘𝐼))
13		pweq 4569	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → 𝒫 𝑖 = 𝒫 𝐼)
14	13	rabeqdv 3415	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘} = {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})
15	14	adantr 480	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘} = {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})
16	12, 15	imaeq12d 6021	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘}) = ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘}))
17	10, 16	fveq12d 6842	. . . . 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 5814	. . . 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 5193	. . 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 481	. 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 3465	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
23		esplyval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
24	23	elexd 3465	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
25		nn0ex 12412	. . . 4 ⊢ ℕ0 ∈ V
26	25	mptex 7172	. . 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 7513	1 ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3400  Vcvv 3441  𝒫 cpw 4555   class class class wbr 5099   ↦ cmpt 5180   “ cima 5628   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363   ↑_m cmap 8768   finSupp cfsupp 9269  0cc0 11031  ℕ₀cn0 12406  ♯chash 14258  ℤRHomczrh 21459  𝟭cind 32932  eSymPolycesply 33725

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-1cn 11089  ax-addcl 11091

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-nn 12151  df-n0 12407  df-esply 33727

This theorem is referenced by:  esplyfval  33732  esplyfval0  33733