Theorem elaa2 47517

Description: Elementhood in the set of nonzero algebraic numbers: when 𝐴 is nonzero, the polynomial 𝑓 can be chosen with a nonzero constant term. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 1-Oct-2020.)

Assertion

Ref	Expression
elaa2	⊢ (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))

Distinct variable group:   𝐴,𝑓

Proof of Theorem elaa2

Dummy variables 𝑔 𝑘 𝑧 𝑗 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aasscn 26698	. . . 4 ⊢ 𝔸 ⊆ ℂ
2		eldifi 4070	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
3	1, 2	sselid 3911	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4		elaa 26696	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
5	2, 4	sylib 221	. . . . 5 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
6	5	simprd 501	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0)
7	2	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ∈ 𝔸)
8		eldifsni 4755	. . . . . . 7 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
9	8	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ≠ 0)
10		eldifi 4070	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11	10	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12		eldifsni 4755	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13	12	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ≠ 0𝑝)
14		simp3 1156	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → (𝑔‘𝐴) = 0)
15		fveq2 6916	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
16	15	neeq1d 3018	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
17	16	cbvrabv 3426	. . . . . . 7 ⊢ {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
18	17	infeq1i 9508	. . . . . 6 ⊢ inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19		fvoveq1 7474	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((coeff‘𝑔)‘(𝑗 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ))) = ((coeff‘𝑔)‘(𝑘 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ))))
20	19	cbvmptv 5215	. . . . . 6 ⊢ (𝑗 ∈ ℕ0 ↦ ((coeff‘𝑔)‘(𝑗 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < )))) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝑔)‘(𝑘 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ))))
21		eqid 2763	. . . . . 6 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝑔) − inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < )))(((𝑗 ∈ ℕ0 ↦ ((coeff‘𝑔)‘(𝑗 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ))))‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝑔) − inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < )))(((𝑗 ∈ ℕ0 ↦ ((coeff‘𝑔)‘(𝑗 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ))))‘𝑘) · (𝑧↑𝑘)))
22	7, 9, 11, 13, 14, 18, 20, 21	elaa2lem 47516	. . . . 5 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
23	22	rexlimdv3a 3171	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))
24	6, 23	mpd 16	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
25	3, 24	jca 521	. 2 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))
26		simpl 488	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
27		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28		coe0 26630	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
29	27, 28	eqtrdi 2814	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
30	29	fveq1d 6918	. . . . . . . . . . . . 13 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31		0nn0 12661	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
32		fvconst2g 7239	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
33	31, 31, 32	mp2an 705	. . . . . . . . . . . . 13 ⊢ ((ℕ0 × {0})‘0) = 0
34	30, 33	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
35	34	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
36		neneq 2965	. . . . . . . . . . . 12 ⊢ (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
37	36	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
38	35, 37	pm2.65da 829	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39		velsn 4600	. . . . . . . . . 10 ⊢ (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
40	38, 39	sylnibr 332	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
41	26, 40	eldifd 4053	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
42	41	adantrr 730	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43		simprr 785	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
44	42, 43	jca 521	. . . . . 6 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0))
45	44	reximi2 3099	. . . . 5 ⊢ (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
46	45	anim2i 629	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
47		elaa 26696	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
48	46, 47	sylibr 237	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ 𝔸)
49		simpr 490	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
50		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑓 𝐴 ∈ ℂ
51		nfre1 3291	. . . . . 6 ⊢ Ⅎ𝑓∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)
52	50, 51	nfan 1934	. . . . 5 ⊢ Ⅎ𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
53		nfv 1949	. . . . 5 ⊢ Ⅎ𝑓 ¬ 𝐴 ∈ {0}
54		simpl3r 1248	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓‘𝐴) = 0)
55		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 0 → (𝑓‘𝐴) = (𝑓‘0))
56		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
57	56	coefv0 26622	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
58	55, 57	sylan9eqr 2820	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
59	58	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
60		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
61	59, 60	eqnetrd 3026	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) ≠ 0)
62	61	neneqd 2964	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
63	62	adantlrr 734	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
64	63	3adantl1 1185	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
65	54, 64	pm2.65da 829	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 = 0)
66		elsng 4598	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
67	66	biimpa 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68	67	3ad2antl1 1204	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
69	65, 68	mtand 828	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70	69	3exp 1137	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0})))
71	70	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0})))
72	52, 53, 71	rexlimd 3273	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0}))
73	49, 72	mpd 16	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
74	48, 73	eldifd 4053	. 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
75	25, 74	impbii 212	1 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  {crab 3416   ∖ cdif 4048  {csn 4588   ↦ cmpt 5192   × cxp 5656  ‘cfv 6566  (class class class)co 7451  infcinf 9470  ℂcc 11237  ℝcr 11238  0cc0 11239   + caddc 11242   · cmul 11244   < clt 11382   − cmin 11580  ℕ₀cn0 12646  ℤcz 12733  ...cfz 13679  ↑cexp 14244  Σcsu 15897  0_𝑝c0p 26045  Polycply 26557  coeffccoe 26559  degcdgr 26560  𝔸caa 26694

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-rlim 15700  df-sum 15898  df-0p 26046  df-ply 26561  df-coe 26563  df-dgr 26564  df-aa 26695

This theorem is used by:  etransc  47566