Theorem esplyfval3 33731

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 2737	. . . . . . . . 9 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
3	2	zrhrhm 21501	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅))
4		zringbas 21443	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤring)
5		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	4, 5	rhmf 20455	. . . . . . . 8 ⊢ ((ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅) → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
7	1, 3, 6	3syl 18	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝑅):ℤ⟶(Base‘𝑅))
8	7	ffnd 6663	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝑅) Fn ℤ)
9		esplyfval3.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
10		ovex 7393	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
11	9, 10	rabex2 5278	. . . . . . . . 9 ⊢ 𝐷 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐷 ∈ V)
13		esplyfval3.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐼 ∈ Fin)
15	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝑅 ∈ Ring)
16		esplyfval3.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ ℕ0)
18	9, 14, 15, 17	esplylem 33725	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
19		indf 12156	. . . . . . . 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 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
21		0zd 12527	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 0 ∈ ℤ)
22		1zzd 12549	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → 1 ∈ ℤ)
23	21, 22	prssd 4766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → {0, 1} ⊆ ℤ)
24	20, 23	fssd 6679	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ)
25		fnfco 6699	. . . . . 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 686	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))) Fn 𝐷)
27	9, 14, 15, 17	esplyfval 33722	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
28	27	fneq1d 6585	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) Fn 𝐷)
30		dffn5 6892	. . . 4 ⊢ (((𝐼eSymPoly𝑅)‘𝐾) Fn 𝐷 ↔ ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓)))
31	29, 30	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓)))
32		eqeq2 2749	. . . . . 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 2749	. . . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐼 ∈ Fin)
35	34	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
36	15	ad2antrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑅 ∈ Ring)
37		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
38		simplr 769	. . . . . . 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 484	. . . . . . 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 33728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((♯‘(𝑓 supp 0)) = 𝐾, 1 , 0 ))
43	27	ad2antrr 727	. . . . . . . 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 6836	. . . . . . 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 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶ℤ)
46		simplr 769	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ 𝐷)
47	45, 46	fvco3d 6934	. . . . . . 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 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
49		simpr 484	. . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 4551	. . . . . . . . . . . . . . . 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 12156	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
57	50, 55, 56	syl2anc 585	. . . . . . . . . . . . . . 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 6645	. . . . . . . . . . . . . 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 32939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
60		f1of 6774	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
62	61	ffnd 6663	. . . . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → 𝑓:𝐼⟶{0, 1})
67	66	frnd 6670	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝑓 ⊆ {0, 1})
68	67	stoic1a 1774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ¬ 𝑓 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
69	46, 68	eldifd 3901	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑓 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
70		ind0 12160	. . . . . . . . . 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 1469	. . . . . . . . 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 6838	. . . . . . . 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 21503	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
74	1, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
75	74	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
76	72, 75	eqtrd 2772	. . . . . . 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 2776	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = 0 )
78	32, 33, 42, 77	ifbothda 4506	. . . . 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 4521	. . . . 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 2790	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝑓) = if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ))
81	80	mpteq2dva 5179	. . 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 2772	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
83		eqid 2737	. . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
84	9	psrbasfsupp 33687	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
85		eqid 2737	. . . . 5 ⊢ (0g‘(𝐼 mPoly 𝑅)) = (0g‘(𝐼 mPoly 𝑅))
86	1	ringgrpd 20214	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
87	83, 84, 39, 85, 13, 86	mpl0 21994	. . . 4 ⊢ (𝜑 → (0g‘(𝐼 mPoly 𝑅)) = (𝐷 × { 0 }))
88	87	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (0g‘(𝐼 mPoly 𝑅)) = (𝐷 × { 0 }))
89	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐼 ∈ Fin)
90	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝑅 ∈ Ring)
91	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ ℕ0)
92		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ¬ 𝐾 ∈ (0...(♯‘𝐼)))
93	91, 92	eldifd 3901	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))
94	9, 89, 90, 93, 85	esplyfval2 33724	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (0g‘(𝐼 mPoly 𝑅)))
95		breq1 5089	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑓 → (ℎ finSupp 0 ↔ 𝑓 finSupp 0))
96	9	eleq2i 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ 𝐷 ↔ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
97	96	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐷 → 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
98	97	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
99	95, 98	elrabrd 32583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 finSupp 0)
100	99	fsuppimpd 9275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ∈ Fin)
101		hashcl 14309	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 0) ∈ Fin → (♯‘(𝑓 supp 0)) ∈ ℕ0)
102	100, 101	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℕ0)
103	102	nn0red 12490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℝ)
104	103	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ∈ ℝ)
105		hashcl 14309	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Fin → (♯‘𝐼) ∈ ℕ0)
106	13, 105	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ0)
107	106	nn0red 12490	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
108	107	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℝ)
109	16	nn0red 12490	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
110	109	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐾 ∈ ℝ)
111		suppssdm 8120	. . . . . . . . . . . . 13 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
112	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝐼 ∈ Fin)
113		nn0ex 12434	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ∈ V
114	113	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → ℕ0 ∈ V)
115	9	ssrab3 4023	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 ⊆ (ℕ0 ↑m 𝐼)
116	115	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ⊆ (ℕ0 ↑m 𝐼))
117	116	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
118	112, 114, 117	elmaprd 32768	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → 𝑓:𝐼⟶ℕ0)
119	111, 118	fssdm 6681	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (𝑓 supp 0) ⊆ 𝐼)
120		hashss 14362	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ (𝑓 supp 0) ⊆ 𝐼) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
121	13, 119, 120	syl2an2r 686	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
122	121	adantlr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≤ (♯‘𝐼))
123	106	nn0zd 12540	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
124	123	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) ∈ ℤ)
125		nn0diffz0 32882	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐼) ∈ ℕ0 → (ℕ0 ∖ (0...(♯‘𝐼))) = (ℤ≥‘((♯‘𝐼) + 1)))
126	89, 105, 125	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (ℕ0 ∖ (0...(♯‘𝐼))) = (ℤ≥‘((♯‘𝐼) + 1)))
127	93, 126	eleqtrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1)))
128	127	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1)))
129		eluzp1l 12806	. . . . . . . . . . 11 ⊢ (((♯‘𝐼) ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘((♯‘𝐼) + 1))) → (♯‘𝐼) < 𝐾)
130	124, 128, 129	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘𝐼) < 𝐾)
131	104, 108, 110, 122, 130	lelttrd 11295	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) < 𝐾)
132	104, 131	ltned 11273	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → (♯‘(𝑓 supp 0)) ≠ 𝐾)
133	132	neneqd 2938	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → ¬ (♯‘(𝑓 supp 0)) = 𝐾)
134	133	intnand 488	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾))
135	134	iffalsed 4478	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) ∧ 𝑓 ∈ 𝐷) → if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 ) = 0 )
136	135	mpteq2dva 5179	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ 0 ))
137		fconstmpt 5686	. . . 4 ⊢ (𝐷 × { 0 }) = (𝑓 ∈ 𝐷 ↦ 0 )
138	136, 137	eqtr4di 2790	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )) = (𝐷 × { 0 }))
139	88, 94, 138	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...(♯‘𝐼))) → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
140	82, 139	pm2.61dan 813	1 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ifcif 4467  𝒫 cpw 4542  {csn 4568  {cpr 4570   class class class wbr 5086   ↦ cmpt 5167   × cxp 5622  ran crn 5625   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360   supp csupp 8103   ↑_m cmap 8766  Fincfn 8886   finSupp cfsupp 9267  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171  𝟭cind 12150  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  ♯chash 14283  Basecbs 17170  0_gc0g 17393  1_rcur 20153  Ringcrg 20205   RingHom crh 20440  ℤ_ringczring 21436  ℤRHomczrh 21489   mPoly cmpl 21896  eSymPolycesply 33715

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-addf 11108  ax-mulf 11109

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-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-ind 12151  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-xnn0 12502  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-0g 17395  df-prds 17401  df-pws 17403  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-rhm 20443  df-subrng 20514  df-subrg 20538  df-cnfld 21345  df-zring 21437  df-zrh 21493  df-psr 21899  df-mpl 21901  df-esply 33717

This theorem is referenced by:  esplyfval1  33732  esplyfvaln  33733  esplyind  33734