Theorem esplyfv 34067

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 2774	. . 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 2774	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
6		esplyfv.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
8		esplyfv.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ (0...(♯‘𝐼)))
10		esplyfv.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
12		esplyfv.0	. . . 4 ⊢ 0 = (0g‘𝑅)
13		esplyfv.1	. . . 4 ⊢ 1 = (1r‘𝑅)
14		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → ran 𝐹 ⊆ {0, 1})
15	3, 5, 7, 9, 11, 12, 13, 14	esplyfv1 34066	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
16	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐼 ∈ Fin)
17	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝑅 ∈ Ring)
18		elfznn0 13677	. . . . . . . 8 ⊢ (𝐾 ∈ (0...(♯‘𝐼)) → 𝐾 ∈ ℕ0)
19	8, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
20	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐾 ∈ ℕ0)
21	3, 16, 17, 20	esplyfval 34060	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
22	21	fveq1d 6884	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝐹))
23		ovex 7449	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
24	3, 23	rabex2 5309	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐷 ∈ V)
26	3, 16, 17, 20	esplylem 34063	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
27		indf 12251	. . . . . 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 596	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})):𝐷⟶{0, 1})
29	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ 𝐷)
30	28, 29	fvco3d 6983	. . . 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 490	. . . . . . . . . . . 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 745	. . . . . . . . . . . . 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 4031	. . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 4569	. . . . . . . . . . . . 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 12251	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑑 ⊆ 𝐼) → ((𝟭‘𝐼)‘𝑑):𝐼⟶{0, 1})
39	32, 37, 38	syl2anc 596	. . . . . . . . . . . 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 6689	. . . . . . . . . . 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 6715	. . . . . . . . . 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 33297	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
43		f1of 6821	. . . . . . . . . . . . . 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 6707	. . . . . . . . . . . 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 6955	. . . . . . . . . . 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 482	. . . . . . . . . 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 3172	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ran 𝐹 ⊆ {0, 1})
50		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) ∧ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})) → ¬ ran 𝐹 ⊆ {0, 1})
51	49, 50	pm2.65da 829	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ¬ 𝐹 ∈ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))
52	29, 51	eldifd 3913	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
53		ind0 12255	. . . . . . 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 1495	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹) = 0)
55	54	fveq2d 6886	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = ((ℤRHom‘𝑅)‘0))
56		eqid 2762	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
57	56, 12	zrh0 21727	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((ℤRHom‘𝑅)‘0) = 0 )
58	6, 57	syl 18	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝑅)‘0) = 0 )
59	58	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘0) = 0 )
60	55, 59	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = 0 )
61	22, 30, 60	3eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = 0 )
62	1, 2, 15, 61	ifbothda 4524	. 2 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if(ran 𝐹 ⊆ {0, 1}, if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ), 0 ))
63		ifan 4539	. 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 2815	1 ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  {crab 3414  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ifcif 4485  𝒫 cpw 4560  {cpr 4589   class class class wbr 5107  ran crn 5660   “ cima 5662   ∘ ccom 5663  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416   supp csupp 8161   ↑_m cmap 8829  Fincfn 8955   finSupp cfsupp 9334  0cc0 11127  1c1 11128  𝟭cind 12245  ℕ₀cn0 12531  ...cfz 13563  ♯chash 14396  0_gc0g 17528  1_rcur 20321  Ringcrg 20373  ℤRHomczrh 21713  eSymPolycesply 34053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-addf 11206  ax-mulf 11207

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-ind 12246  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-seq 14068  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-starv 17361  df-tset 17365  df-ple 17366  df-ds 17368  df-unif 17369  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-grp 19061  df-minusg 19062  df-mulg 19192  df-subg 19247  df-ghm 19342  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-cring 20376  df-rhm 20614  df-subrng 20709  df-subrg 20733  df-cnfld 21587  df-zring 21661  df-zrh 21717  df-esply 34055

This theorem is used by:  esplysply  34068