Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elaa2 Structured version   Visualization version   GIF version

Theorem elaa2 40972
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 24292 . . . 4 𝔸 ⊆ ℂ
2 eldifi 3875 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ 𝔸)
31, 2sseldi 3742 . . 3 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ∈ ℂ)
4 elaa 24290 . . . . . 6 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
52, 4sylib 208 . . . . 5 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0))
65simprd 482 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0)
723ad2ant1 1128 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ∈ 𝔸)
8 eldifsni 4466 . . . . . . 7 (𝐴 ∈ (𝔸 ∖ {0}) → 𝐴 ≠ 0)
983ad2ant1 1128 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝐴 ≠ 0)
10 eldifi 3875 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ∈ (Poly‘ℤ))
11103ad2ant2 1129 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ∈ (Poly‘ℤ))
12 eldifsni 4466 . . . . . . 7 (𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑔 ≠ 0𝑝)
13123ad2ant2 1129 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → 𝑔 ≠ 0𝑝)
14 simp3 1133 . . . . . 6 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → (𝑔𝐴) = 0)
15 fveq2 6353 . . . . . . . . 9 (𝑚 = 𝑛 → ((coeff‘𝑔)‘𝑚) = ((coeff‘𝑔)‘𝑛))
1615neeq1d 2991 . . . . . . . 8 (𝑚 = 𝑛 → (((coeff‘𝑔)‘𝑚) ≠ 0 ↔ ((coeff‘𝑔)‘𝑛) ≠ 0))
1716cbvrabv 3339 . . . . . . 7 {𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0} = {𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}
1817infeq1i 8551 . . . . . 6 inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < ) = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑛) ≠ 0}, ℝ, < )
19 oveq1 6821 . . . . . . . 8 (𝑗 = 𝑘 → (𝑗 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < )) = (𝑘 + inf({𝑚 ∈ ℕ0 ∣ ((coeff‘𝑔)‘𝑚) ≠ 0}, ℝ, < )))
2019fveq2d 6357 . . . . . . 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, < ))))
2120cbvmptv 4902 . . . . . 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, < ))))
22 eqid 2760 . . . . . 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)))
237, 9, 11, 13, 14, 18, 21, 22elaa2lem 40971 . . . . 5 ((𝐴 ∈ (𝔸 ∖ {0}) ∧ 𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔𝐴) = 0) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
2423rexlimdv3a 3171 . . . 4 (𝐴 ∈ (𝔸 ∖ {0}) → (∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔𝐴) = 0 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)))
256, 24mpd 15 . . 3 (𝐴 ∈ (𝔸 ∖ {0}) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
263, 25jca 555 . 2 (𝐴 ∈ (𝔸 ∖ {0}) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)))
27 simpl 474 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ (Poly‘ℤ))
28 fveq2 6353 . . . . . . . . . . . . . . 15 (𝑓 = 0𝑝 → (coeff‘𝑓) = (coeff‘0𝑝))
29 coe0 24231 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
3028, 29syl6eq 2810 . . . . . . . . . . . . . 14 (𝑓 = 0𝑝 → (coeff‘𝑓) = (ℕ0 × {0}))
3130fveq1d 6355 . . . . . . . . . . . . 13 (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = ((ℕ0 × {0})‘0))
32 0nn0 11519 . . . . . . . . . . . . . 14 0 ∈ ℕ0
33 fvconst2g 6632 . . . . . . . . . . . . . 14 ((0 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((ℕ0 × {0})‘0) = 0)
3432, 32, 33mp2an 710 . . . . . . . . . . . . 13 ((ℕ0 × {0})‘0) = 0
3531, 34syl6eq 2810 . . . . . . . . . . . 12 (𝑓 = 0𝑝 → ((coeff‘𝑓)‘0) = 0)
3635adantl 473 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ((coeff‘𝑓)‘0) = 0)
37 neneq 2938 . . . . . . . . . . . 12 (((coeff‘𝑓)‘0) ≠ 0 → ¬ ((coeff‘𝑓)‘0) = 0)
3837ad2antlr 765 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝑓 = 0𝑝) → ¬ ((coeff‘𝑓)‘0) = 0)
3936, 38pm2.65da 601 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 = 0𝑝)
40 velsn 4337 . . . . . . . . . 10 (𝑓 ∈ {0𝑝} ↔ 𝑓 = 0𝑝)
4139, 40sylnibr 318 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → ¬ 𝑓 ∈ {0𝑝})
4227, 41eldifd 3726 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
4342adantrr 755 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
44 simprr 813 . . . . . . 7 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓𝐴) = 0)
4543, 44jca 555 . . . . . 6 ((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓𝐴) = 0))
4645reximi2 3148 . . . . 5 (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0)
4746anim2i 594 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
48 elaa 24290 . . . 4 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))
4947, 48sylibr 224 . . 3 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ 𝔸)
50 simpr 479 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
51 nfv 1992 . . . . . 6 𝑓 𝐴 ∈ ℂ
52 nfre1 3143 . . . . . 6 𝑓𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)
5351, 52nfan 1977 . . . . 5 𝑓(𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
54 nfv 1992 . . . . 5 𝑓 ¬ 𝐴 ∈ {0}
55 simpl3r 1289 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → (𝑓𝐴) = 0)
56 fveq2 6353 . . . . . . . . . . . . . . 15 (𝐴 = 0 → (𝑓𝐴) = (𝑓‘0))
57 eqid 2760 . . . . . . . . . . . . . . . 16 (coeff‘𝑓) = (coeff‘𝑓)
5857coefv0 24223 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℤ) → (𝑓‘0) = ((coeff‘𝑓)‘0))
5956, 58sylan9eqr 2816 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
6059adantlr 753 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) = ((coeff‘𝑓)‘0))
61 simplr 809 . . . . . . . . . . . . 13 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ((coeff‘𝑓)‘0) ≠ 0)
6260, 61eqnetrd 2999 . . . . . . . . . . . 12 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → (𝑓𝐴) ≠ 0)
6362neneqd 2937 . . . . . . . . . . 11 (((𝑓 ∈ (Poly‘ℤ) ∧ ((coeff‘𝑓)‘0) ≠ 0) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6463adantlrr 759 . . . . . . . . . 10 (((𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
65643adantl1 1172 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 = 0) → ¬ (𝑓𝐴) = 0)
6655, 65pm2.65da 601 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 = 0)
67 elsng 4335 . . . . . . . . . 10 (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
6867biimpa 502 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
69683ad2antl1 1201 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) ∧ 𝐴 ∈ {0}) → 𝐴 = 0)
7066, 69mtand 694 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑓 ∈ (Poly‘ℤ) ∧ (((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 ∈ {0})
71703exp 1113 . . . . . 6 (𝐴 ∈ ℂ → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0})))
7271adantr 472 . . . . 5 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0})))
7353, 54, 72rexlimd 3164 . . . 4 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → (∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) → ¬ 𝐴 ∈ {0}))
7450, 73mpd 15 . . 3 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → ¬ 𝐴 ∈ {0})
7549, 74eldifd 3726 . 2 ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0)) → 𝐴 ∈ (𝔸 ∖ {0}))
7626, 75impbii 199 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  {crab 3054  cdif 3712  {csn 4321  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6814  infcinf 8514  cc 10146  cr 10147  0cc0 10148   + caddc 10151   · cmul 10153   < clt 10286  cmin 10478  0cn0 11504  cz 11589  ...cfz 12539  cexp 13074  Σcsu 14635  0𝑝c0p 23655  Polycply 24159  coeffccoe 24161  degcdgr 24162  𝔸caa 24288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-0p 23656  df-ply 24163  df-coe 24165  df-dgr 24166  df-aa 24289
This theorem is referenced by:  etransc  41021
  Copyright terms: Public domain W3C validator