Theorem esplyfv 33926

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 2773	. . 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 2773	. . 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 485	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
6		esplyfv.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
8		esplyfv.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
10		esplyfv.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
11	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
12		esplyfv.0	. . . 4 ⊢ 0 = (0g‘𝑅)
13		esplyfv.1	. . . 4 ⊢ 1 = (1r‘𝑅)
14		simpr 489	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ⊆ {0, 1})
15	3, 5, 7, 9, 11, 12, 13, 14	esplyfv1 33925	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
16	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
17	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
18		elfznn0 13647	. . . . . . . 8 ⊢ (𝐾 ∈ (0...(♯‘𝐼)) → 𝐾 ∈ ℕ0)
19	8, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
20	19	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ ℕ0)
21	3, 16, 17, 20	esplyfval 33919	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
22	21	fveq1d 6883	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝐹))
23		ovex 7443	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
24	3, 23	rabex2 5311	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐷 ∈ V)
26	3, 16, 17, 20	esplylem 33922	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
27		indf 12223	. . . . . 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 595	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
29	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
30	28, 29	fvco3d 6982	. . . 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 489	. . . . . . . . . . . 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 744	. . . . . . . . . . . . 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 4033	. . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 4570	. . . . . . . . . . . . 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 12223	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
39	32, 37, 38	syl2anc 595	. . . . . . . . . . . 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 6688	. . . . . . . . . . 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 6714	. . . . . . . . . 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 33150	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
43		f1of 6820	. . . . . . . . . . . . . 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 19	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (𝟭‘𝐼):𝒫 𝐼⟶({0, 1} ↑m 𝐼))
45	44	ffnd 6706	. . . . . . . . . . . 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 6954	. . . . . . . . . . 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 481	. . . . . . . . . 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 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ¬ ran 𝐹 ⊆ {0, 1})
51	49, 50	pm2.65da 828	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ¬ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
52	29, 51	eldifd 3915	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
53		ind0 12227	. . . . . . 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 1493	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹) = 0)
55	54	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = ((ℤRHom‘𝑅)‘0))
56		eqid 2761	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
57	56, 12	zrh0 21642	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
58	6, 57	syl 18	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
59	58	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
60	55, 59	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = 0 )
61	22, 30, 60	3eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = 0 )
62	1, 2, 15, 61	ifbothda 4525	. 2 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if(ran 𝐹 ⊆ {0, 1}, if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ), 0 ))
63		ifan 4540	. 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 2814	1 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  {crab 3414  Vcvv 3453   ∖ cdif 3901   ⊆ wss 3904  ifcif 4486  𝒫 cpw 4561  {cpr 4590   class class class wbr 5108  ran crn 5662   “ cima 5664   ∘ ccom 5665  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   supp csupp 8155   ↑_m cmap 8823  Fincfn 8942   finSupp cfsupp 9320  0cc0 11099  1c1 11100  𝟭cind 12217  ℕ₀cn0 12503  ...cfz 13534  ♯chash 14365  0_gc0g 17491  1_rcur 20262  Ringcrg 20314  ℤRHomczrh 21628  eSymPolycesply 33912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-addf 11178  ax-mulf 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-ind 12218  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-seq 14037  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-grp 19002  df-minusg 19003  df-mulg 19133  df-subg 19188  df-ghm 19283  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-rhm 20553  df-subrng 20630  df-subrg 20654  df-cnfld 21502  df-zring 21576  df-zrh 21632  df-esply 33914

This theorem is referenced by:  esplysply  33927