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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 ∧ (𝑓𝐴) = 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.
StepHypRef Expression
1 aasscn 24510 . . . 4 𝔸 ⊆ ℂ
2 eldifi 3955 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
31, 2sseldi 3819 . . 3 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4 elaa 24508 . . . . . 6 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
52, 4sylib 210 . . . . 5 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
65simprd 491 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0)
723ad2ant1 1124 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ∈ 𝔸)
8 eldifsni 4553 . . . . . . 7 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
983ad2ant1 1124 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ≠ 0)
10 eldifi 3955 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11103ad2ant2 1125 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12 eldifsni 4553 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13123ad2ant2 1125 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ≠ 0𝑝)
14 simp3 1129 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → (𝑔𝐴) = 0)
15 fveq2 6446 . . . . . . . . 9 (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
1615neeq1d 3028 . . . . . . . 8 (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
1716cbvrabv 3396 . . . . . . 7 {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
1817infeq1i 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, < ))))
2019cbvmptv 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) · (𝑧𝑘))) = (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) · (𝑧𝑘)))
227, 9, 11, 13, 14, 18, 20, 21elaa2lem 41377 . . . . 5 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
2322rexlimdv3a 3215 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → (∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)))
246, 23mpd 15 . . 3 (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
253, 24jca 507 . 2 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 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})
2927, 28syl6eq 2830 . . . . . . . . . . . . . 14 (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
3029fveq1d 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)
3331, 31, 32mp2an 682 . . . . . . . . . . . . 13 ((ℕ0 × {0})‘0) = 0
3430, 33syl6eq 2830 . . . . . . . . . . . 12 (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
3534adantl 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)
3736ad2antlr 717 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
3835, 37pm2.65da 807 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39 velsn 4414 . . . . . . . . . 10 (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
4038, 39sylnibr 321 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
4126, 40eldifd 3803 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
4241adantrr 707 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43 simprr 763 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓𝐴) = 0)
4442, 43jca 507 . . . . . 6 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓𝐴) = 0))
4544reximi2 3191 . . . . 5 (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0)
4645anim2i 610 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
47 elaa 24508 . . . 4 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
4846, 47sylibr 226 . . 3 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ 𝔸)
49 simpr 479 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
50 nfv 1957 . . . . . 6 𝑓 𝐴 ∈ ℂ
51 nfre1 3186 . . . . . 6 𝑓𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)
5250, 51nfan 1946 . . . . 5 𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
53 nfv 1957 . . . . 5 𝑓 ¬ 𝐴 ∈ {0}
54 simpl3r 1260 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓𝐴) = 0)
55 fveq2 6446 . . . . . . . . . . . . . . 15 (𝐴 = 0 → (𝑓𝐴) = (𝑓‘0))
56 eqid 2778 . . . . . . . . . . . . . . . 16 (coeff‘𝑓) = (coeff‘𝑓)
5756coefv0 24441 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
5855, 57sylan9eqr 2836 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
5958adantlr 705 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
60 simplr 759 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
6159, 60eqnetrd 3036 . . . . . . . . . . . 12 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) ≠ 0)
6261neneqd 2974 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6362adantlrr 711 . . . . . . . . . 10 (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
64633adantl1 1168 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6554, 64pm2.65da 807 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 = 0)
66 elsng 4412 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
6766biimpa 470 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68673ad2antl1 1193 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
6965, 68mtand 806 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70693exp 1109 . . . . . 6 (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0})))
7170adantr 474 . . . . 5 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0})))
7252, 53, 71rexlimd 3208 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0}))
7349, 72mpd 15 . . 3 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 ∈ {0})
7448, 73eldifd 3803 . 2 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
7525, 74impbii 201 1 (𝐴 ∈ (𝔸 ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)))
Colors of variables: wff setvar class
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  cc 10270  cr 10271  0cc0 10272   + caddc 10275   · cmul 10277   < clt 10411  cmin 10606  0cn0 11642  cz 11728  ...cfz 12643  cexp 13178  Σcsu 14824  0𝑝c0p 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
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