Theorem esplyfval3 33940

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 2763	. . . . . . . . 9 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 21642	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅))
4		zringbas 21584	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤring)
5		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	4, 5	rhmf 20567	. . . . . . . 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 6708	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝑅) Fn ℤ)
9		esplyfval3.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
10		ovex 7445	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
11	9, 10	rabex2 5313	. . . . . . . . 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 33934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
19		indf 12225	. . . . . . . 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 12604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 0 ∈ ℤ)
22		1zzd 12626	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 1 ∈ ℤ)
23	21, 22	prssd 4789	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → {0, 1} ⊆ ℤ)
24	20, 23	fssd 6725	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ)
25		fnfco 6745	. . . . . 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 33931	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
28	27	fneq1d 6630	. . . . 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 6941	. . . 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 2775	. . . . . 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 2775	. . . . . 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 33937	. . . . . 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 6885	. . . . . . 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 6984	. . . . . . 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 4035	. . . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . 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 12225	. . . . . . . . . . . . . . . 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 6690	. . . . . . . . . . . . . 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 33162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
60		f1of 6822	. . . . . . . . . . . . . . . . . 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 6708	. . . . . . . . . . . . . . . 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 6956	. . . . . . . . . . . . . . 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 6716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝑓 ⊆ {0, 1})
68	67	stoic1a 1802	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ¬ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
69	46, 68	eldifd 3917	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
70		ind0 12229	. . . . . . . . . 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 1495	. . . . . . . . 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 6887	. . . . . . . 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 21644	. . . . . . . . . 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 2798	. . . . . . 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 2802	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = 0 )
78	32, 33, 42, 77	ifbothda 4527	. . . . 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 4542	. . . . 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 2816	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ))
81	80	mpteq2dva 5205	. . 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 2798	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
83		eqid 2763	. . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
84	9	psrbasfsupp 33879	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
85		eqid 2763	. . . . 5 ⊢ (0g‘(𝐼 mPoly 𝑅)) = (0g‘(𝐼 mPoly 𝑅))
86	1	ringgrpd 20325	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
87	83, 84, 39, 85, 13, 86	mpl0 22136	. . . 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 3917	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))
94	9, 89, 90, 93, 85	esplyfval2 33933	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (0g‘(𝐼 mPoly 𝑅)))
95		breq1 5113	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑓 → (ℎ finSupp 0 ↔ 𝑓 finSupp 0))
96	9	eleq2i 2855	. . . . . . . . . . . . . . 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 3654	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 finSupp 0)
99	98	fsuppimpd 9330	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ∈ Fin)
100		hashcl 14394	. . . . . . . . . . . 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 12567	. . . . . . . . . 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 14394	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Fin → (♯‘𝐼) ∈ ℕ0)
105	13, 104	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ0)
106	105	nn0red 12567	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
107	106	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℝ)
108	16	nn0red 12567	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
109	108	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐾 ∈ ℝ)
110		suppssdm 8174	. . . . . . . . . . . . 13 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
111	13	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝐼 ∈ Fin)
112		nn0ex 12511	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
113	112	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → ℕ0 ∈ V)
114	9	ssrab3 4037	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 ⊆ (ℕ0 ↑m 𝐼)
115	114	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ0 ↑m 𝐼))
116	115	sselda 3938	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
117	111, 113, 116	elmaprd 33003	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓:𝐼⟶ℕ0)
118	110, 117	fssdm 6727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ⊆ 𝐼)
119		hashss 14447	. . . . . . . . . . . 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 12617	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
123	122	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℤ)
124		nn0diffz0 33117	. . . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 12890	. . . . . . . . . . 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 11369	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) < 𝐾)
131	103, 130	ltned 11347	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≠ 𝐾)
132	131	neneqd 2963	. . . . . . 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 4499	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ) = 0 )
135	134	mpteq2dva 5205	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ 0 ))
136		fconstmpt 5725	. . . 4 ⊢ (𝐷 × { 0 }) = (𝑓 ∈ 𝐷 ↦ 0 )
137	135, 136	eqtr4di 2816	. . 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 2808	. 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 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ifcif 4488  𝒫 cpw 4563  {csn 4590  {cpr 4592   class class class wbr 5110   ↦ cmpt 5193   × cxp 5661  ran crn 5664   “ cima 5666   ∘ ccom 5667   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412   supp csupp 8157   ↑_m cmap 8825  Fincfn 8944   finSupp cfsupp 9322  ℝcr 11100  0cc0 11101  1c1 11102   + caddc 11104   < clt 11244   ≤ cle 11245  𝟭cind 12219  ℕ₀cn0 12505  ℤcz 12592  ℤ_≥cuz 12863  ...cfz 13536  ♯chash 14368  Basecbs 17270  0_gc0g 17493  1_rcur 20264  Ringcrg 20316   RingHom crh 20552  ℤ_ringczring 21577  ℤRHomczrh 21630   mPoly cmpl 22037  eSymPolycesply 33924

This theorem was proved from 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-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-addf 11180  ax-mulf 11181

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  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-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-ind 12220  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-xnn0 12579  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-0g 17495  df-prds 17501  df-pws 17503  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-minusg 19005  df-mulg 19135  df-subg 19190  df-ghm 19285  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-cring 20319  df-rhm 20555  df-subrng 20632  df-subrg 20656  df-cnfld 21504  df-zring 21578  df-zrh 21634  df-psr 22040  df-mpl 22042  df-esply 33926

This theorem is referenced by:  esplyfval1  33941  esplyfvaln  33942  esplyind  33943