Theorem esplyfv 33719

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 2749	. . 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 2749	. . 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 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
6		esplyfv.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
8		esplyfv.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
10		esplyfv.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
12		esplyfv.0	. . . 4 ⊢ 0 = (0g‘𝑅)
13		esplyfv.1	. . . 4 ⊢ 1 = (1r‘𝑅)
14		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ⊆ {0, 1})
15	3, 5, 7, 9, 11, 12, 13, 14	esplyfv1 33718	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
16	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
17	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
18		elfznn0 13537	. . . . . . . 8 ⊢ (𝐾 ∈ (0...(♯‘𝐼)) → 𝐾 ∈ ℕ0)
19	8, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ ℕ0)
21	3, 16, 17, 20	esplyfval 33712	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
22	21	fveq1d 6834	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝐹))
23		ovex 7391	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
24	3, 23	rabex2 5276	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐷 ∈ V)
26	3, 16, 17, 20	esplylem 33715	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
27		indf 32917	. . . . . 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 585	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
29	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
30	28, 29	fvco3d 6932	. . . 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 484	. . . . . . . . . . . 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 733	. . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 4551	. . . . . . . . . . . . 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 32917	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
39	32, 37, 38	syl2anc 585	. . . . . . . . . . . 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 6643	. . . . . . . . . . 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 6668	. . . . . . . . . 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 32929	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
43		f1of 6772	. . . . . . . . . . . . . 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 6661	. . . . . . . . . . . 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 6905	. . . . . . . . . . 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 476	. . . . . . . . . 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 3146	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝐹 ⊆ {0, 1})
50		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ¬ ran 𝐹 ⊆ {0, 1})
51	49, 50	pm2.65da 817	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ¬ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
52	29, 51	eldifd 3901	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
53		ind0 32920	. . . . . . 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 1469	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹) = 0)
55	54	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = ((ℤRHom‘𝑅)‘0))
56		eqid 2737	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
57	56, 12	zrh0 21470	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
58	6, 57	syl 17	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
60	55, 59	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = 0 )
61	22, 30, 60	3eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = 0 )
62	1, 2, 15, 61	ifbothda 4506	. 2 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if(ran 𝐹 ⊆ {0, 1}, if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ), 0 ))
63		ifan 4521	. 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 2790	1 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = 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  {cpr 4570   class class class wbr 5086  ran crn 5623   “ cima 5625   ∘ ccom 5626  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7358   supp csupp 8101   ↑_m cmap 8764  Fincfn 8884   finSupp cfsupp 9265  0cc0 11027  1c1 11028  ℕ₀cn0 12402  ...cfz 13424  ♯chash 14254  0_gc0g 17360  1_rcur 20120  Ringcrg 20172  ℤRHomczrh 21456  𝟭cind 32912  eSymPolycesply 33705

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 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-addf 11106  ax-mulf 11107

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-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12609  df-uz 12753  df-fz 13425  df-seq 13926  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-starv 17193  df-tset 17197  df-ple 17198  df-ds 17200  df-unif 17201  df-0g 17362  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18709  df-grp 18870  df-minusg 18871  df-mulg 19002  df-subg 19057  df-ghm 19146  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-rhm 20410  df-subrng 20481  df-subrg 20505  df-cnfld 21312  df-zring 21404  df-zrh 21460  df-ind 32913  df-esply 33707

This theorem is referenced by:  esplysply  33720