Theorem elaa2 46478

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 26282	. . . 4 ⊢ 𝔸 ⊆ ℂ
2		eldifi 4083	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
3	1, 2	sselid 3931	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4		elaa 26280	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
5	2, 4	sylib 218	. . . . 5 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
6	5	simprd 495	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0)
7	2	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ∈ 𝔸)
8		eldifsni 4746	. . . . . . 7 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
9	8	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ≠ 0)
10		eldifi 4083	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11	10	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12		eldifsni 4746	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13	12	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ≠ 0𝑝)
14		simp3 1138	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → (𝑔‘𝐴) = 0)
15		fveq2 6834	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
16	15	neeq1d 2991	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
17	16	cbvrabv 3409	. . . . . . 7 ⊢ {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
18	17	infeq1i 9382	. . . . . 6 ⊢ inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19		fvoveq1 7381	. . . . . . 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 5202	. . . . . 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 2736	. . . . . 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 46477	. . . . 5 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
23	22	rexlimdv3a 3141	. . . 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 511	. 2 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))
26		simpl 482	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
27		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28		coe0 26217	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
29	27, 28	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
30	29	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31		0nn0 12416	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
32		fvconst2g 7148	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
33	31, 31, 32	mp2an 692	. . . . . . . . . . . . 13 ⊢ ((ℕ0 × {0})‘0) = 0
34	30, 33	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
35	34	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
36		neneq 2938	. . . . . . . . . . . 12 ⊢ (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
37	36	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
38	35, 37	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39		velsn 4596	. . . . . . . . . 10 ⊢ (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
40	38, 39	sylnibr 329	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
41	26, 40	eldifd 3912	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
42	41	adantrr 717	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43		simprr 772	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
44	42, 43	jca 511	. . . . . 6 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0))
45	44	reximi2 3069	. . . . 5 ⊢ (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
46	45	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
47		elaa 26280	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
48	46, 47	sylibr 234	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ 𝔸)
49		simpr 484	. . . 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 1915	. . . . . 6 ⊢ Ⅎ𝑓 𝐴 ∈ ℂ
51		nfre1 3261	. . . . . 6 ⊢ Ⅎ𝑓∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)
52	50, 51	nfan 1900	. . . . 5 ⊢ Ⅎ𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
53		nfv 1915	. . . . 5 ⊢ Ⅎ𝑓 ¬ 𝐴 ∈ {0}
54		simpl3r 1230	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓‘𝐴) = 0)
55		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 0 → (𝑓‘𝐴) = (𝑓‘0))
56		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
57	56	coefv0 26209	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
58	55, 57	sylan9eqr 2793	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
59	58	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
60		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
61	59, 60	eqnetrd 2999	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) ≠ 0)
62	61	neneqd 2937	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
63	62	adantlrr 721	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
64	63	3adantl1 1167	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
65	54, 64	pm2.65da 816	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 = 0)
66		elsng 4594	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
67	66	biimpa 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68	67	3ad2antl1 1186	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
69	65, 68	mtand 815	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70	69	3exp 1119	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0})))
71	70	adantr 480	. . . . 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 3243	. . . 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 3912	. 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
75	25, 74	impbii 209	1 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399   ∖ cdif 3898  {csn 4580   ↦ cmpt 5179   × cxp 5622  ‘cfv 6492  (class class class)co 7358  infcinf 9344  ℂcc 11024  ℝcr 11025  0cc0 11026   + caddc 11029   · cmul 11031   < clt 11166   − cmin 11364  ℕ₀cn0 12401  ℤcz 12488  ...cfz 13423  ↑cexp 13984  Σcsu 15609  0_𝑝c0p 25626  Polycply 26145  coeffccoe 26147  degcdgr 26148  𝔸caa 26278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-rlim 15412  df-sum 15610  df-0p 25627  df-ply 26149  df-coe 26151  df-dgr 26152  df-aa 26279

This theorem is referenced by:  etransc  46527