Theorem esplyfval3 33879

Description: Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (Contributed by Thierry Arnoux, 25-Jan-2026.)

Hypotheses

Ref	Expression
esplyfval3.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
esplyfval3.i	⊢ (𝜑 → 𝐼 ∈ Fin)
esplyfval3.r	⊢ (𝜑 → 𝑅 ∈ Ring)
esplyfval3.k	⊢ (𝜑 → 𝐾 ∈ ℕ0)
esplyfval3.1	⊢ 0 = (0g‘𝑅)
esplyfval3.2	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
esplyfval3	⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))

Distinct variable groups:   0 ,𝑓   𝐷,𝑓   ℎ,𝐼,𝑓   𝑓,𝐾   𝑅,𝑓   𝜑,𝑓

Allowed substitution hints:   𝜑(ℎ)   𝐷(ℎ)   𝑅(ℎ)   1 (𝑓,ℎ)   𝐾(ℎ)   0 (ℎ)

Proof of Theorem esplyfval3

Dummy variables 𝑑 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esplyfval3.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		eqid 2765	. . . . . . . . 9 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 21621	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅))
4		zringbas 21563	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤring)
5		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	4, 5	rhmf 20557	. . . . . . . 8 ⊢ ((ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
7	1, 3, 6	3syl 19	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
8	7	ffnd 6696	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝑅) Fn ℤ)
9		esplyfval3.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
10		ovex 7433	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
11	9, 10	rabex2 5302	. . . . . . . . 9 ⊢ 𝐷 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐷 ∈ V)
13		esplyfval3.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
14	13	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐼 ∈ Fin)
15	1	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝑅 ∈ Ring)
16		esplyfval3.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ ℕ0)
18	9, 14, 15, 17	esplylem 33873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
19		indf 12215	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
20	12, 18, 19	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
21		0zd 12594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 0 ∈ ℤ)
22		1zzd 12616	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 1 ∈ ℤ)
23	21, 22	prssd 4783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → {0, 1} ⊆ ℤ)
24	20, 23	fssd 6713	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ)
25		fnfco 6733	. . . . . 6 ⊢ (((ℤRHom‘𝑅) Fn ℤ ∧ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ) → ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))) Fn 𝐷)
26	8, 24, 25	syl2an2r 697	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))) Fn 𝐷)
27	9, 14, 15, 17	esplyfval 33870	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
28	27	fneq1d 6618	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → (((𝐼eSymPoly𝑅)‘𝐾) Fn 𝐷 ↔ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))) Fn 𝐷))
29	26, 28	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) Fn 𝐷)
30		dffn5 6929	. . . 4 ⊢ (((𝐼eSymPoly𝑅)‘𝐾) Fn 𝐷 ↔ ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓)))
31	29, 30	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓)))
32		eqeq2 2777	. . . . . 6 ⊢ (if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ) = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 ) → ((((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ) ↔ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 )))
33		eqeq2 2777	. . . . . 6 ⊢ ( 0 = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 ) → ((((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = 0 ↔ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 )))
34	14	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐼 ∈ Fin)
35	34	adantr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
36	15	ad2antrr 738	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑅 ∈ Ring)
37		simpllr 787	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
38		simplr 780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ 𝐷)
39		esplyfval3.1	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
40		esplyfval3.2	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
41		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → ran 𝑓 ⊆ {0, 1})
42	9, 35, 36, 37, 38, 39, 40, 41	esplyfv1 33876	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ))
43	27	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
44	43	fveq1d 6873	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝑓))
45	24	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ)
46		simplr 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ 𝐷)
47	45, 46	fvco3d 6972	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝑓) = ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝑓)))
48	18	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
49		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → ((𝟭‘𝐼)‘𝑑) = 𝑓)
50	34	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → 𝐼 ∈ Fin)
51		ssrab2 4036	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼
52	51	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼)
53	52	sselda 3939	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) → 𝑑 ∈ 𝒫 𝐼)
54	53	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → 𝑑 ∈ 𝒫 𝐼)
55	54	elpwid 4567	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → 𝑑 ⊆ 𝐼)
56		indf 12215	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
57	50, 55, 56	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
58	49, 57	feq1dd 6678	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝑓) → 𝑓:𝐼⟶{0, 1})
59		indf1o 33097	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
60		f1of 6810	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
61	34, 59, 60	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
62	61	ffnd 6696	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (𝟭‘𝐼) Fn 𝒫 𝐼)
63	51	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼)
64	62, 63	fvelimabd 6944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ↔ ∃𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ((𝟭‘𝐼)‘𝑑) = 𝑓))
65	64	biimpa 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ∃𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ((𝟭‘𝐼)‘𝑑) = 𝑓)
66	58, 65	r19.29a 3173	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → 𝑓:𝐼⟶{0, 1})
67	66	frnd 6704	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝑓 ⊆ {0, 1})
68	67	stoic1a 1795	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ¬ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
69	46, 68	eldifd 3918	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
70		ind0 12219	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷 ∧ 𝑓 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝑓) = 0)
71	11, 48, 69, 70	mp3an2i 1490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝑓) = 0)
72	71	fveq2d 6875	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝑓)) = ((ℤRHom‘𝑅)‘0))
73	2, 39	zrh0 21623	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
74	1, 73	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
75	74	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
76	72, 75	eqtrd 2800	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝑓)) = 0 )
77	44, 47, 76	3eqtrd 2804	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = 0 )
78	32, 33, 42, 77	ifbothda 4522	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 ))
79		ifan 4537	. . . . 5 ⊢ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ) = if(ran 𝑓 ⊆ {0, 1}, if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ), 0 )
80	78, 79	eqtr4di 2818	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ))
81	80	mpteq2dva 5198	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓)) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
82	31, 81	eqtrd 2800	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
83		eqid 2765	. . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
84	9	psrbasfsupp 33818	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
85		eqid 2765	. . . . 5 ⊢ (0g‘(𝐼 mPoly 𝑅)) = (0g‘(𝐼 mPoly 𝑅))
86	1	ringgrpd 20315	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
87	83, 84, 39, 85, 13, 86	mpl0 22115	. . . 4 ⊢ (𝜑 → (0g‘(𝐼 mPoly 𝑅)) = (𝐷 × { 0 }))
88	87	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (0g‘(𝐼 mPoly 𝑅)) = (𝐷 × { 0 }))
89	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐼 ∈ Fin)
90	1	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝑅 ∈ Ring)
91	16	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ ℕ0)
92		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ¬ 𝐾 ∈ (0...(♯‘𝐼)))
93	91, 92	eldifd 3918	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))
94	9, 89, 90, 93, 85	esplyfval2 33872	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (0g‘(𝐼 mPoly 𝑅)))
95		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑓 → (ℎ finSupp 0 ↔ 𝑓 finSupp 0))
96	9	eleq2i 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐷 ↔ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
97	96	bilani 509	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
98	95, 97	elrabrd 3656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 finSupp 0)
99	98	fsuppimpd 9317	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ∈ Fin)
100		hashcl 14383	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 0) ∈ Fin → (♯‘(𝑓 supp 0)) ∈ ℕ0)
101	99, 100	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℕ0)
102	101	nn0red 12557	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℝ)
103	102	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℝ)
104		hashcl 14383	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Fin → (♯‘𝐼) ∈ ℕ0)
105	13, 104	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ0)
106	105	nn0red 12557	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
107	106	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℝ)
108	16	nn0red 12557	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
109	108	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐾 ∈ ℝ)
110		suppssdm 8161	. . . . . . . . . . . . 13 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
111	13	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝐼 ∈ Fin)
112		nn0ex 12501	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
113	112	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → ℕ0 ∈ V)
114	9	ssrab3 4038	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 ⊆ (ℕ0 ↑m 𝐼)
115	114	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ0 ↑m 𝐼))
116	115	sselda 3939	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
117	111, 113, 116	elmaprd 32937	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓:𝐼⟶ℕ0)
118	110, 117	fssdm 6715	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ⊆ 𝐼)
119		hashss 14436	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ (𝑓 supp 0) ⊆ 𝐼) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
120	13, 118, 119	syl2an2r 697	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
121	120	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
122	105	nn0zd 12607	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
123	122	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℤ)
124		nn0diffz0 33051	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐼) ∈ ℕ0 → (ℕ0 ∖ (0...(♯‘𝐼))) = (ℤ≥‘((♯‘𝐼) + 1)))
125	89, 104, 124	3syl 19	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (ℕ0 ∖ (0...(♯‘𝐼))) = (ℤ≥‘((♯‘𝐼) + 1)))
126	93, 125	eleqtrd 2867	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1)))
127	126	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1)))
128		eluzp1l 12880	. . . . . . . . . . 11 ⊢ (((♯‘𝐼) ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1))) → (♯‘𝐼) < 𝐾)
129	123, 127, 128	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) < 𝐾)
130	103, 107, 109, 121, 129	lelttrd 11356	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) < 𝐾)
131	103, 130	ltned 11334	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≠ 𝐾)
132	131	neneqd 2965	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → ¬ (♯‘(𝑓 supp 0)) = 𝐾)
133	132	intnand 493	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾))
134	133	iffalsed 4494	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ) = 0 )
135	134	mpteq2dva 5198	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ 0 ))
136		fconstmpt 5714	. . . 4 ⊢ (𝐷 × { 0 }) = (𝑓 ∈ 𝐷 ↦ 0 )
137	135, 136	eqtr4di 2818	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )) = (𝐷 × { 0 }))
138	88, 94, 137	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
139	82, 138	pm2.61dan 824	1 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  𝒫 cpw 4558  {csn 4585  {cpr 4587   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650  ran crn 5653   “ cima 5655   ∘ ccom 5656   Fn wfn 6520  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   supp csupp 8144   ↑_m cmap 8812  Fincfn 8931   finSupp cfsupp 9309  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   < clt 11231   ≤ cle 11232  𝟭cind 12209  ℕ₀cn0 12495  ℤcz 12582  ℤ_≥cuz 12853  ...cfz 13526  ♯chash 14357  Basecbs 17259  0_gc0g 17482  1_rcur 20254  Ringcrg 20306   RingHom crh 20542  ℤ_ringczring 21556  ℤRHomczrh 21609   mPoly cmpl 22016  eSymPolycesply 33863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-addf 11167  ax-mulf 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-ind 12210  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-starv 17315  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-unif 17323  df-hom 17324  df-cco 17325  df-0g 17484  df-prds 17490  df-pws 17492  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-grp 18993  df-minusg 18994  df-mulg 19125  df-subg 19180  df-ghm 19275  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-rhm 20545  df-subrng 20622  df-subrg 20646  df-cnfld 21483  df-zring 21557  df-zrh 21613  df-psr 22019  df-mpl 22021  df-esply 33865

This theorem is referenced by:  esplyfval1  33880  esplyfvaln  33881  esplyind  33882