Theorem elaa2 46746

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 26348	. . . 4 ⊢ 𝔸 ⊆ ℂ
2		eldifi 4075	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
3	1, 2	sselid 3925	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4		elaa 26346	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
5	2, 4	sylib 220	. . . . 5 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
6	5	simprd 498	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0)
7	2	3ad2ant1 1142	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ∈ 𝔸)
8		eldifsni 4740	. . . . . . 7 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
9	8	3ad2ant1 1142	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ≠ 0)
10		eldifi 4075	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11	10	3ad2ant2 1143	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12		eldifsni 4740	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13	12	3ad2ant2 1143	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ≠ 0𝑝)
14		simp3 1147	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → (𝑔‘𝐴) = 0)
15		fveq2 6852	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
16	15	neeq1d 3006	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
17	16	cbvrabv 3414	. . . . . . 7 ⊢ {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
18	17	infeq1i 9411	. . . . . 6 ⊢ inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19		fvoveq1 7404	. . . . . . 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 5194	. . . . . 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 2752	. . . . . 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 46745	. . . . 5 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
23	22	rexlimdv3a 3157	. . . 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 15	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
25	3, 24	jca 518	. 2 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))
26		simpl 485	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
27		fveq2 6852	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28		coe0 26285	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
29	27, 28	eqtrdi 2803	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
30	29	fveq1d 6854	. . . . . . . . . . . . 13 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31		0nn0 12482	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
32		fvconst2g 7171	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
33	31, 31, 32	mp2an 700	. . . . . . . . . . . . 13 ⊢ ((ℕ0 × {0})‘0) = 0
34	30, 33	eqtrdi 2803	. . . . . . . . . . . 12 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
35	34	adantl 484	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
36		neneq 2953	. . . . . . . . . . . 12 ⊢ (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
37	36	ad2antlr 735	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
38	35, 37	pm2.65da 824	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39		velsn 4588	. . . . . . . . . 10 ⊢ (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
40	38, 39	sylnibr 331	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
41	26, 40	eldifd 3906	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
42	41	adantrr 725	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43		simprr 780	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
44	42, 43	jca 518	. . . . . 6 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0))
45	44	reximi2 3085	. . . . 5 ⊢ (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
46	45	anim2i 625	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
47		elaa 26346	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
48	46, 47	sylibr 236	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ 𝔸)
49		simpr 487	. . . 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 1924	. . . . . 6 ⊢ Ⅎ𝑓 𝐴 ∈ ℂ
51		nfre1 3277	. . . . . 6 ⊢ Ⅎ𝑓∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)
52	50, 51	nfan 1909	. . . . 5 ⊢ Ⅎ𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
53		nfv 1924	. . . . 5 ⊢ Ⅎ𝑓 ¬ 𝐴 ∈ {0}
54		simpl3r 1239	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓‘𝐴) = 0)
55		fveq2 6852	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 0 → (𝑓‘𝐴) = (𝑓‘0))
56		eqid 2752	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
57	56	coefv0 26277	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
58	55, 57	sylan9eqr 2809	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
59	58	adantlr 723	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
60		simplr 776	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
61	59, 60	eqnetrd 3014	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) ≠ 0)
62	61	neneqd 2952	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
63	62	adantlrr 729	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
64	63	3adantl1 1176	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
65	54, 64	pm2.65da 824	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 = 0)
66		elsng 4586	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
67	66	biimpa 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68	67	3ad2antl1 1195	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
69	65, 68	mtand 823	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70	69	3exp 1128	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0})))
71	70	adantr 483	. . . . 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 3259	. . . 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 15	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
74	48, 73	eldifd 3906	. 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
75	25, 74	impbii 211	1 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  {crab 3404   ∖ cdif 3892  {csn 4572   ↦ cmpt 5171   × cxp 5634  ‘cfv 6506  (class class class)co 7381  infcinf 9373  ℂcc 11057  ℝcr 11058  0cc0 11059   + caddc 11062   · cmul 11064   < clt 11202   − cmin 11400  ℕ₀cn0 12467  ℤcz 12554  ...cfz 13498  ↑cexp 14060  Σcsu 15685  0_𝑝c0p 25700  Polycply 26213  coeffccoe 26215  degcdgr 26216  𝔸caa 26344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-of 7645  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-pm 8795  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-sup 9374  df-inf 9375  df-oi 9444  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-rp 12980  df-fz 13499  df-fzo 13646  df-fl 13788  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-rlim 15488  df-sum 15686  df-0p 25701  df-ply 26217  df-coe 26219  df-dgr 26220  df-aa 26345

This theorem is referenced by:  etransc  46795