Theorem elaa2 41378

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 24510	. . . 4 ⊢ 𝔸 ⊆ ℂ
2		eldifi 3955	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
3	1, 2	sseldi 3819	. . 3 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4		elaa 24508	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
5	2, 4	sylib 210	. . . . 5 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0))
6	5	simprd 491	. . . 4 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘𝐴) = 0)
7	2	3ad2ant1 1124	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ∈ 𝔸)
8		eldifsni 4553	. . . . . . 7 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
9	8	3ad2ant1 1124	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝐴 ≠ 0)
10		eldifi 3955	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11	10	3ad2ant2 1125	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12		eldifsni 4553	. . . . . . 7 ⊢ (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13	12	3ad2ant2 1125	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → 𝑔 ≠ 0𝑝)
14		simp3 1129	. . . . . 6 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → (𝑔‘𝐴) = 0)
15		fveq2 6446	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
16	15	neeq1d 3028	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
17	16	cbvrabv 3396	. . . . . . 7 ⊢ {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
18	17	infeq1i 8672	. . . . . 6 ⊢ inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19		fvoveq1 6945	. . . . . . 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 4985	. . . . . 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 2778	. . . . . 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 41377	. . . . 5 ⊢ ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔‘𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
23	22	rexlimdv3a 3215	. . . 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 507	. 2 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))
26		simpl 476	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
27		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28		coe0 24449	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
29	27, 28	syl6eq 2830	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
30	29	fveq1d 6448	. . . . . . . . . . . . 13 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31		0nn0 11659	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
32		fvconst2g 6739	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
33	31, 31, 32	mp2an 682	. . . . . . . . . . . . 13 ⊢ ((ℕ0 × {0})‘0) = 0
34	30, 33	syl6eq 2830	. . . . . . . . . . . 12 ⊢ (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
35	34	adantl 475	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
36		neneq 2975	. . . . . . . . . . . 12 ⊢ (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
37	36	ad2antlr 717	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
38	35, 37	pm2.65da 807	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39		velsn 4414	. . . . . . . . . 10 ⊢ (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
40	38, 39	sylnibr 321	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
41	26, 40	eldifd 3803	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
42	41	adantrr 707	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43		simprr 763	. . . . . . 7 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
44	42, 43	jca 507	. . . . . 6 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0))
45	44	reximi2 3191	. . . . 5 ⊢ (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
46	45	anim2i 610	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
47		elaa 24508	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
48	46, 47	sylibr 226	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ 𝔸)
49		simpr 479	. . . 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 1957	. . . . . 6 ⊢ Ⅎ𝑓 𝐴 ∈ ℂ
51		nfre1 3186	. . . . . 6 ⊢ Ⅎ𝑓∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)
52	50, 51	nfan 1946	. . . . 5 ⊢ Ⅎ𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
53		nfv 1957	. . . . 5 ⊢ Ⅎ𝑓 ¬ 𝐴 ∈ {0}
54		simpl3r 1260	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓‘𝐴) = 0)
55		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = 0 → (𝑓‘𝐴) = (𝑓‘0))
56		eqid 2778	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
57	56	coefv0 24441	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
58	55, 57	sylan9eqr 2836	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
59	58	adantlr 705	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) = ((coeff‘𝑓)‘0))
60		simplr 759	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
61	59, 60	eqnetrd 3036	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓‘𝐴) ≠ 0)
62	61	neneqd 2974	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
63	62	adantlrr 711	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
64	63	3adantl1 1168	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓‘𝐴) = 0)
65	54, 64	pm2.65da 807	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 = 0)
66		elsng 4412	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
67	66	biimpa 470	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68	67	3ad2antl1 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
69	65, 68	mtand 806	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70	69	3exp 1109	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) → ¬ 𝐴 ∈ {0})))
71	70	adantr 474	. . . . 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 3208	. . . 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 3803	. 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
75	25, 74	impbii 201	1 ⊢ (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091  {crab 3094   ∖ cdif 3789  {csn 4398   ↦ cmpt 4965   × cxp 5353  ‘cfv 6135  (class class class)co 6922  infcinf 8635  ℂcc 10270  ℝcr 10271  0cc0 10272   + caddc 10275   · cmul 10277   < clt 10411   − cmin 10606  ℕ₀cn0 11642  ℤcz 11728  ...cfz 12643  ↑cexp 13178  Σcsu 14824  0_𝑝c0p 23873  Polycply 24377  coeffccoe 24379  degcdgr 24380  𝔸caa 24506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-rlim 14628  df-sum 14825  df-0p 23874  df-ply 24381  df-coe 24383  df-dgr 24384  df-aa 24507

This theorem is referenced by:  etransc  41427