Theorem esplyfv 33869

Description: Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.)

Hypotheses

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

Assertion

Ref	Expression
esplyfv	⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))

Distinct variable group:   ℎ,𝐼

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

Proof of Theorem esplyfv

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

Step	Hyp	Ref	Expression
1		eqeq2 2775	. . 3 ⊢ (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 )))
2		eqeq2 2775	. . 3 ⊢ ( 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 )))
3		esplyfv.d	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
4		esplyfv.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin)
5	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
6		esplyfv.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
8		esplyfv.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
10		esplyfv.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
11	10	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
12		esplyfv.0	. . . 4 ⊢ 0 = (0g‘𝑅)
13		esplyfv.1	. . . 4 ⊢ 1 = (1r‘𝑅)
14		simpr 488	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ⊆ {0, 1})
15	3, 5, 7, 9, 11, 12, 13, 14	esplyfv1 33868	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
16	4	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
17	6	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
18		elfznn0 13635	. . . . . . . 8 ⊢ (𝐾 ∈ (0...(♯‘𝐼)) → 𝐾 ∈ ℕ0)
19	8, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ ℕ0)
21	3, 16, 17, 20	esplyfval 33862	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
22	21	fveq1d 6869	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝐹))
23		ovex 7429	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
24	3, 23	rabex2 5298	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐷 ∈ V)
26	3, 16, 17, 20	esplylem 33865	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
27		indf 12211	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
28	25, 26, 27	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
29	10	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
30	28, 29	fvco3d 6968	. . . 4 ⊢ ((𝜑 ∧ ¬ 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)))
31		simpr 488	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → ((𝟭‘𝐼)‘𝑑) = 𝐹)
32	4	ad4antr 742	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → 𝐼 ∈ Fin)
33		ssrab2 4034	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼
34	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼)
35	34	sselda 3937	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) → 𝑑 ∈ 𝒫 𝐼)
36	35	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → 𝑑 ∈ 𝒫 𝐼)
37	36	elpwid 4565	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → 𝑑 ⊆ 𝐼)
38		indf 12211	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
39	32, 37, 38	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
40	31, 39	feq1dd 6674	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → 𝐹:𝐼⟶{0, 1})
41	40	frnd 6700	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) ∧ 𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ∧ ((𝟭‘𝐼)‘𝑑) = 𝐹) → ran 𝐹 ⊆ {0, 1})
42		indf1o 33048	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
43		f1of 6806	. . . . . . . . . . . . . 14 ⊢ ((𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
44	16, 42, 43	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
45	44	ffnd 6692	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (𝟭‘𝐼) Fn 𝒫 𝐼)
46	33	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ⊆ 𝒫 𝐼)
47	45, 46	fvelimabd 6940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ↔ ∃𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ((𝟭‘𝐼)‘𝑑) = 𝐹))
48	47	biimpa 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ∃𝑑 ∈ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾} ((𝟭‘𝐼)‘𝑑) = 𝐹)
49	41, 48	r19.29a 3171	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝐹 ⊆ {0, 1})
50		simplr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ¬ ran 𝐹 ⊆ {0, 1})
51	49, 50	pm2.65da 826	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ¬ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
52	29, 51	eldifd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
53		ind0 12215	. . . . . . 7 ⊢ ((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)
54	24, 26, 52, 53	mp3an2i 1488	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹) = 0)
55	54	fveq2d 6871	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = ((ℤRHom‘𝑅)‘0))
56		eqid 2763	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
57	56, 12	zrh0 21572	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
58	6, 57	syl 17	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
59	58	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
60	55, 59	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = 0 )
61	22, 30, 60	3eqtrd 2802	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = 0 )
62	1, 2, 15, 61	ifbothda 4520	. 2 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if(ran 𝐹 ⊆ {0, 1}, if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ), 0 ))
63		ifan 4535	. 2 ⊢ if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ) = if(ran 𝐹 ⊆ {0, 1}, if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ), 0 )
64	62, 63	eqtr4di 2816	1 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∃wrex 3087  {crab 3415  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ifcif 4481  𝒫 cpw 4556  {cpr 4585   class class class wbr 5101  ran crn 5649   “ cima 5651   ∘ ccom 5652  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396   supp csupp 8140   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9305  0cc0 11084  1c1 11085  𝟭cind 12205  ℕ₀cn0 12491  ...cfz 13522  ♯chash 14353  0_gc0g 17478  1_rcur 20241  Ringcrg 20293  ℤRHomczrh 21558  eSymPolycesply 33855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161  ax-addf 11163  ax-mulf 11164

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-ind 12206  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-seq 14025  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-starv 17311  df-tset 17315  df-ple 17316  df-ds 17318  df-unif 17319  df-0g 17480  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-grp 18988  df-minusg 18989  df-mulg 19120  df-subg 19175  df-ghm 19264  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-ring 20295  df-cring 20296  df-rhm 20531  df-subrng 20606  df-subrg 20630  df-cnfld 21432  df-zring 21506  df-zrh 21562  df-esply 33857

This theorem is referenced by:  esplysply  33870