Theorem esplyfv 34244

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 2772	. . 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 2772	. . 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 34243	. . 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 13754	. . . . . . . 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 34237	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
22	21	fveq1d 6887	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = (((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))‘𝐹))
23		ovex 7453	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
24	3, 23	rabex2 5301	. . . . . . 7 ⊢ 𝐷 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐷 ∈ V)
26	3, 16, 17, 20	esplylem 34240	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
27		indf 12326	. . . . . 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 6986	. . . 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 4005	. . . . . . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . 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 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 12326	. . . . . . . . . . . . 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 6692	. . . . . . . . . . 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 6718	. . . . . . . . . 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 33473	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → (𝟭‘𝐼):𝒫 𝐼–1-1-onto→({0, 1} ↑m 𝐼))
43		f1of 6824	. . . . . . . . . . . . . 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 6710	. . . . . . . . . . . 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 6958	. . . . . . . . . . 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 3170	. . . . . . . . 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 4050	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → 𝐹 ∈ (𝐷 ∖ ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾})))
53		ind0 12330	. . . . . . 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 6889	. . . . 5 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = ((ℤRHom‘𝑅)‘0))
56		eqid 2760	. . . . . . . 8 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
57	56, 12	zrh0 21819	. . . . . . 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 2795	. . . 4 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {0, 1}) → ((ℤRHom‘𝑅)‘(((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))‘𝐹)) = 0 )
61	22, 30, 60	3eqtrd 2799	. . 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 2813	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 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045  ifcif 4481  𝒫 cpw 4556  {cpr 4585   class class class wbr 5102  ran crn 5651   “ cima 5653   ∘ ccom 5654  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   supp csupp 8177   ↑_m cmap 8847  Fincfn 8973   finSupp cfsupp 9353  0cc0 11200  1c1 11201  𝟭cind 12320  ℕ₀cn0 12606  ...cfz 13639  ♯chash 14474  0_gc0g 17610  1_rcur 20407  Ringcrg 20459  ℤRHomczrh 21805  eSymPolycesply 34230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-addf 11279  ax-mulf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-ind 12321  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-seq 14145  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-grp 19147  df-minusg 19148  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-rhm 20702  df-subrng 20798  df-subrg 20822  df-cnfld 21679  df-zring 21753  df-zrh 21809  df-esply 34232

This theorem is used by:  esplysply  34245