Theorem elaa2 46215

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 26224	. . . 4 ⊢ 𝔸 ⊆ ℂ
2		eldifi 4082	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
3	1, 2	sselid 3933	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4		elaa 26222	. . . . . 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 4741	. . . . . . 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 4082	. . . . . . 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 4741	. . . . . . 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 6822	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
16	15	neeq1d 2984	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
17	16	cbvrabv 3405	. . . . . . 7 ⊢ {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
18	17	infeq1i 9369	. . . . . 6 ⊢ inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19		fvoveq1 7372	. . . . . . 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 5196	. . . . . 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 2729	. . . . . 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 46214	. . . . 5 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
23	22	rexlimdv3a 3134	. . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28		coe0 26159	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
29	27, 28	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
30	29	fveq1d 6824	. . . . . . . . . . . . 13 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31		0nn0 12399	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
32		fvconst2g 7138	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
33	31, 31, 32	mp2an 692	. . . . . . . . . . . . 13 ⊢ ((ℕ0 × {0})‘0) = 0
34	30, 33	eqtrdi 2780	. . . . . . . . . . . 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 2931	. . . . . . . . . . . 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 4593	. . . . . . . . . 10 ⊢ (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
40	38, 39	sylnibr 329	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
41	26, 40	eldifd 3914	. . . . . . . 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 3062	. . . . 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 26222	. . . 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 1914	. . . . . 6 ⊢ Ⅎ𝑓 𝐴 ∈ ℂ
51		nfre1 3254	. . . . . 6 ⊢ Ⅎ𝑓∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)
52	50, 51	nfan 1899	. . . . 5 ⊢ Ⅎ𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
53		nfv 1914	. . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 0 → (𝑓‘𝐴) = (𝑓‘0))
56		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
57	56	coefv0 26151	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
58	55, 57	sylan9eqr 2786	. . . . . . . . . . . . . 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 2992	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) ≠ 0)
62	61	neneqd 2930	. . . . . . . . . . 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 4591	. . . . . . . . . 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 3236	. . . 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 3914	. 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 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394   ∖ cdif 3900  {csn 4577   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  (class class class)co 7349  infcinf 9331  ℂcc 11007  ℝcr 11008  0cc0 11009   + caddc 11012   · cmul 11014   < clt 11149   − cmin 11347  ℕ₀cn0 12384  ℤcz 12471  ...cfz 13410  ↑cexp 13968  Σcsu 15593  0_𝑝c0p 25568  Polycply 26087  coeffccoe 26089  degcdgr 26090  𝔸caa 26220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-0p 25569  df-ply 26091  df-coe 26093  df-dgr 26094  df-aa 26221

This theorem is referenced by:  etransc  46264