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Theorem elaa2 46190
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 26375 . . . 4 𝔸 ⊆ ℂ
2 eldifi 4141 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
31, 2sselid 3993 . . 3 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4 elaa 26373 . . . . . 6 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
52, 4sylib 218 . . . . 5 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
65simprd 495 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0)
723ad2ant1 1132 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ∈ 𝔸)
8 eldifsni 4795 . . . . . . 7 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
983ad2ant1 1132 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ≠ 0)
10 eldifi 4141 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11103ad2ant2 1133 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12 eldifsni 4795 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13123ad2ant2 1133 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ≠ 0𝑝)
14 simp3 1137 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → (𝑔𝐴) = 0)
15 fveq2 6907 . . . . . . . . 9 (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
1615neeq1d 2998 . . . . . . . 8 (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
1716cbvrabv 3444 . . . . . . 7 {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
1817infeq1i 9516 . . . . . 6 inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19 fvoveq1 7454 . . . . . . 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 5261 . . . . . 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 2735 . . . . . 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 46189 . . . . 5 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
2322rexlimdv3a 3157 . . . 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 511 . 2 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)))
26 simpl 482 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
27 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
28 coe0 26310 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2927, 28eqtrdi 2791 . . . . . . . . . . . . . 14 (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
3029fveq1d 6909 . . . . . . . . . . . . 13 (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
31 0nn0 12539 . . . . . . . . . . . . . 14 0 ∈ ℕ0
32 fvconst2g 7222 . . . . . . . . . . . . . 14 ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
3331, 31, 32mp2an 692 . . . . . . . . . . . . 13 ((ℕ0 × {0})‘0) = 0
3430, 33eqtrdi 2791 . . . . . . . . . . . 12 (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
3534adantl 481 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
36 neneq 2944 . . . . . . . . . . . 12 (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
3736ad2antlr 727 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
3835, 37pm2.65da 817 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
39 velsn 4647 . . . . . . . . . 10 (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
4038, 39sylnibr 329 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
4126, 40eldifd 3974 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
4241adantrr 717 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
43 simprr 773 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓𝐴) = 0)
4442, 43jca 511 . . . . . 6 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓𝐴) = 0))
4544reximi2 3077 . . . . 5 (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0)
4645anim2i 617 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
47 elaa 26373 . . . 4 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
4846, 47sylibr 234 . . 3 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ 𝔸)
49 simpr 484 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
50 nfv 1912 . . . . . 6 𝑓 𝐴 ∈ ℂ
51 nfre1 3283 . . . . . 6 𝑓𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)
5250, 51nfan 1897 . . . . 5 𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
53 nfv 1912 . . . . 5 𝑓 ¬ 𝐴 ∈ {0}
54 simpl3r 1228 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓𝐴) = 0)
55 fveq2 6907 . . . . . . . . . . . . . . 15 (𝐴 = 0 → (𝑓𝐴) = (𝑓‘0))
56 eqid 2735 . . . . . . . . . . . . . . . 16 (coeff‘𝑓) = (coeff‘𝑓)
5756coefv0 26302 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
5855, 57sylan9eqr 2797 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
5958adantlr 715 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
60 simplr 769 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
6159, 60eqnetrd 3006 . . . . . . . . . . . 12 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) ≠ 0)
6261neneqd 2943 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6362adantlrr 721 . . . . . . . . . 10 (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
64633adantl1 1165 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6554, 64pm2.65da 817 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 = 0)
66 elsng 4645 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
6766biimpa 476 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
68673ad2antl1 1184 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
6965, 68mtand 816 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 ∈ {0})
70693exp 1118 . . . . . 6 (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0})))
7170adantr 480 . . . . 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 3264 . . . 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 3974 . 2 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
7525, 74impbii 209 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  cdif 3960  {csn 4631  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  infcinf 9479  cc 11151  cr 11152  0cc0 11153   + caddc 11156   · cmul 11158   < clt 11293  cmin 11490  0cn0 12524  cz 12611  ...cfz 13544  cexp 14099  Σcsu 15719  0𝑝c0p 25718  Polycply 26238  coeffccoe 26240  degcdgr 26241  𝔸caa 26371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-coe 26244  df-dgr 26245  df-aa 26372
This theorem is referenced by:  etransc  46239
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