Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elaa2lem Structured version   Visualization version   GIF version

Theorem elaa2lem 42395
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 42396. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 1-Oct-2020.)
Hypotheses
Ref Expression
elaa2lem.a (𝜑𝐴 ∈ 𝔸)
elaa2lem.an0 (𝜑𝐴 ≠ 0)
elaa2lem.g (𝜑𝐺 ∈ (Poly‘ℤ))
elaa2lem.gn0 (𝜑𝐺 ≠ 0𝑝)
elaa2lem.ga (𝜑 → (𝐺𝐴) = 0)
elaa2lem.m 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
elaa2lem.i 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
elaa2lem.f 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)))
Assertion
Ref Expression
elaa2lem (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
Distinct variable groups:   𝐴,𝑓   𝐴,𝑘,𝑧   𝑓,𝐹   𝑘,𝐺   𝑛,𝐺   𝑧,𝐺   𝑘,𝐼,𝑧   𝑘,𝑀   𝑛,𝑀   𝑧,𝑀   𝜑,𝑘,𝑧
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝐴(𝑛)   𝐹(𝑧,𝑘,𝑛)   𝐺(𝑓)   𝐼(𝑓,𝑛)   𝑀(𝑓)

Proof of Theorem elaa2lem
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 elaa2lem.f . . . 4 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)))
21a1i 11 . . 3 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))))
3 zsscn 11977 . . . . 5 ℤ ⊆ ℂ
43a1i 11 . . . 4 (𝜑 → ℤ ⊆ ℂ)
5 elaa2lem.g . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘ℤ))
6 dgrcl 24750 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
75, 6syl 17 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℕ0)
87nn0zd 12073 . . . . . . 7 (𝜑 → (deg‘𝐺) ∈ ℤ)
9 elaa2lem.m . . . . . . . . 9 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10 ssrab2 4053 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11 nn0uz 12268 . . . . . . . . . . . . 13 0 = (ℤ‘0)
1210, 11sseqtri 4000 . . . . . . . . . . . 12 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0)
1312a1i 11 . . . . . . . . . . 11 (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0))
14 elaa2lem.gn0 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 ≠ 0𝑝)
1514neneqd 3018 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐺 = 0𝑝)
16 eqid 2818 . . . . . . . . . . . . . . . . . 18 (deg‘𝐺) = (deg‘𝐺)
17 eqid 2818 . . . . . . . . . . . . . . . . . 18 (coeff‘𝐺) = (coeff‘𝐺)
1816, 17dgreq0 24782 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ (Poly‘ℤ) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
195, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
2015, 19mtbid 325 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)
2120neqned 3020 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
227, 21jca 512 . . . . . . . . . . . . 13 (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
2423neeq1d 3072 . . . . . . . . . . . . . 14 (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2524elrab 3677 . . . . . . . . . . . . 13 ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2622, 25sylibr 235 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
2726ne0d 4298 . . . . . . . . . . 11 (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
28 infssuzcl 12320 . . . . . . . . . . 11 (({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0) ∧ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
2913, 27, 28syl2anc 584 . . . . . . . . . 10 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
3010, 29sseldi 3962 . . . . . . . . 9 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
319, 30eqeltrid 2914 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3231nn0zd 12073 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
338, 32zsubcld 12080 . . . . . 6 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
349a1i 11 . . . . . . . 8 (𝜑𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
35 infssuzle 12319 . . . . . . . . 9 (({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0) ∧ (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
3613, 26, 35syl2anc 584 . . . . . . . 8 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
3734, 36eqbrtrd 5079 . . . . . . 7 (𝜑𝑀 ≤ (deg‘𝐺))
387nn0red 11944 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℝ)
3931nn0red 11944 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
4038, 39subge0d 11218 . . . . . . 7 (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
4137, 40mpbird 258 . . . . . 6 (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
4233, 41jca 512 . . . . 5 (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
43 elnn0z 11982 . . . . 5 (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
4442, 43sylibr 235 . . . 4 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
45 0zd 11981 . . . . . . . 8 (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
4617coef2 24748 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
475, 45, 46syl2anc2 585 . . . . . . 7 (𝜑 → (coeff‘𝐺):ℕ0⟶ℤ)
4847adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (coeff‘𝐺):ℕ0⟶ℤ)
49 simpr 485 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
5031adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
5149, 50nn0addcld 11947 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5248, 51ffvelrnd 6844 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
53 elaa2lem.i . . . . 5 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
5452, 53fmptd 6870 . . . 4 (𝜑𝐼:ℕ0⟶ℤ)
55 elplyr 24718 . . . 4 ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ0𝐼:ℕ0⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
564, 44, 54, 55syl3anc 1363 . . 3 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
572, 56eqeltrd 2910 . 2 (𝜑𝐹 ∈ (Poly‘ℤ))
58 simpr 485 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
5958iftrued 4471 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
60 iffalse 4472 . . . . . . . . . . 11 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
6160adantl 482 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
62 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6338ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
6439ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
6563, 64resubcld 11056 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
66 nn0re 11894 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
6766ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
6865, 67ltnled 10775 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
6962, 68mpbird 258 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
7063, 64, 67ltsubaddd 11224 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
7169, 70mpbid 233 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) < (𝑘 + 𝑀))
72 olc 862 . . . . . . . . . . . . 13 ((deg‘𝐺) < (𝑘 + 𝑀) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
7371, 72syl 17 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
745ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
7551adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ0)
7616, 17dgrlt 24783 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℤ) ∧ (𝑘 + 𝑀) ∈ ℕ0) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
7774, 75, 76syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
7873, 77mpbid 233 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
7978simprd 496 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
8061, 79eqtr4d 2856 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8159, 80pm2.61dan 809 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8281mpteq2dva 5152 . . . . . . 7 (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
8347, 4fssd 6521 . . . . . . . . . 10 (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
8483adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
85 elfznn0 12988 . . . . . . . . . . 11 (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ0)
8685adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ0)
8731adantr 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ0)
8886, 87nn0addcld 11947 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
8984, 88ffvelrnd 6844 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
90 eqidd 2819 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
91 simpl 483 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝜑)
9253a1i 11 . . . . . . . . . . . . . . 15 (𝜑𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
9392, 52fvmpt2d 6773 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9491, 86, 93syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9594adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9695oveq1d 7160 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
9790, 96sumeq12rdv 15052 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
9897mpteq2dva 5152 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
992, 98eqtrd 2853 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
10057, 44, 89, 99coeeq2 24759 . . . . . . 7 (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
10182, 100, 923eqtr4d 2863 . . . . . 6 (𝜑 → (coeff‘𝐹) = 𝐼)
102101fveq1d 6665 . . . . 5 (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
103 oveq1 7152 . . . . . . . . 9 (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
104103adantl 482 . . . . . . . 8 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
1053, 32sseldi 3962 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
106105addid2d 10829 . . . . . . . . 9 (𝜑 → (0 + 𝑀) = 𝑀)
107106adantr 481 . . . . . . . 8 ((𝜑𝑘 = 0) → (0 + 𝑀) = 𝑀)
108104, 107eqtrd 2853 . . . . . . 7 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
109108fveq2d 6667 . . . . . 6 ((𝜑𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
110 0nn0 11900 . . . . . . 7 0 ∈ ℕ0
111110a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℕ0)
11247, 31ffvelrnd 6844 . . . . . 6 (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
11392, 109, 111, 112fvmptd 6767 . . . . 5 (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
114 eqidd 2819 . . . . 5 (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
115102, 113, 1143eqtrd 2857 . . . 4 (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
11634, 29eqeltrd 2910 . . . . . 6 (𝜑𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
117 fveq2 6663 . . . . . . . 8 (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
118117neeq1d 3072 . . . . . . 7 (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
119118elrab 3677 . . . . . 6 (𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
120116, 119sylib 219 . . . . 5 (𝜑 → (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
121120simprd 496 . . . 4 (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
122115, 121eqnetrd 3080 . . 3 (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
1235, 45syl 17 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
124 aasscn 24834 . . . . . . . . . . 11 𝔸 ⊆ ℂ
125 elaa2lem.a . . . . . . . . . . 11 (𝜑𝐴 ∈ 𝔸)
126124, 125sseldi 3962 . . . . . . . . . 10 (𝜑𝐴 ∈ ℂ)
12791, 126syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
128127, 86expcld 13498 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴𝑘) ∈ ℂ)
12989, 128mulcld 10649 . . . . . . 7 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
130 fvoveq1 7168 . . . . . . . 8 (𝑘 = (𝑗𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)))
131 oveq2 7153 . . . . . . . 8 (𝑘 = (𝑗𝑀) → (𝐴𝑘) = (𝐴↑(𝑗𝑀)))
132130, 131oveq12d 7163 . . . . . . 7 (𝑘 = (𝑗𝑀) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
13332, 123, 33, 129, 132fsumshft 15123 . . . . . 6 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
1343, 8sseldi 3962 . . . . . . . . . 10 (𝜑 → (deg‘𝐺) ∈ ℂ)
135134, 105npcand 10989 . . . . . . . . 9 (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
136106, 135oveq12d 7163 . . . . . . . 8 (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
137136sumeq1d 15046 . . . . . . 7 (𝜑 → Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
138 elfzelz 12896 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
139138adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
1403, 139sseldi 3962 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
141105adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
142140, 141npcand 10989 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗𝑀) + 𝑀) = 𝑗)
143142fveq2d 6667 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
144143oveq1d 7160 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
145126adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
146 elaa2lem.an0 . . . . . . . . . . . . 13 (𝜑𝐴 ≠ 0)
147146adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
14832adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
149145, 147, 148, 139expsubd 13509 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗𝑀)) = ((𝐴𝑗) / (𝐴𝑀)))
150149oveq2d 7161 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
15183adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
152 0red 10632 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
15339adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
154139zred 12075 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
15531nn0ge0d 11946 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ≤ 𝑀)
156155adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
157 elfzle1 12898 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀𝑗)
158157adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀𝑗)
159152, 153, 154, 156, 158letrd 10785 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
160139, 159jca 512 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
161 elnn0z 11982 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
162160, 161sylibr 235 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
163151, 162ffvelrnd 6844 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
164145, 162expcld 13498 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
165126, 31expcld 13498 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑀) ∈ ℂ)
166165adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ∈ ℂ)
167145, 147, 148expne0d 13504 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ≠ 0)
168163, 164, 166, 167divassd 11439 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
169168eqcomd 2824 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
170150, 169eqtr2d 2854 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
171144, 170eqtr4d 2856 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
172171sumeq2dv 15048 . . . . . . 7 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
173137, 172eqtrd 2853 . . . . . 6 (𝜑 → Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
17431, 11eleqtrdi 2920 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘0))
175 fzss1 12934 . . . . . . . 8 (𝑀 ∈ (ℤ‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
176174, 175syl 17 . . . . . . 7 (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
177163, 164mulcld 10649 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
178177, 166, 167divcld 11404 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) ∈ ℂ)
17932ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
1808ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
181 eldifi 4100 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
182 elfznn0 12988 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ0)
183182nn0zd 12073 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℤ)
184181, 183syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
185184ad2antlr 723 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
186179, 180, 1853jca 1120 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ))
187 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
18839ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
189185zred 12075 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
190188, 189lenltd 10774 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
191187, 190mpbird 258 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀𝑗)
192 elfzle2 12899 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
193181, 192syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
194193ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
195186, 191, 194jca32 516 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
196 elfz2 12887 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀...(deg‘𝐺)) ↔ ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
197195, 196sylibr 235 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
198 eldifn 4101 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
199198ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
200197, 199condan 814 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
201200adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 < 𝑀)
2029a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
20312a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0))
204181, 182syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
205204adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ0)
206 neqne 3021 . . . . . . . . . . . . . . . . . . 19 (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
207206adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
208205, 207jca 512 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
209 fveq2 6663 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
210209neeq1d 3072 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
211210elrab 3677 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
212208, 211sylibr 235 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
213212adantll 710 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
214 infssuzle 12319 . . . . . . . . . . . . . . 15 (({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0) ∧ 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
215203, 213, 214syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
216202, 215eqbrtrd 5079 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀𝑗)
21739ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
218184zred 12075 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
219218ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
220217, 219lenltd 10774 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
221216, 220mpbid 233 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
222201, 221condan 814 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
223222oveq1d 7160 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = (0 · (𝐴𝑗)))
224126adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
225204adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ0)
226224, 225expcld 13498 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴𝑗) ∈ ℂ)
227226mul02d 10826 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴𝑗)) = 0)
228223, 227eqtrd 2853 . . . . . . . . 9 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = 0)
229228oveq1d 7160 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
230126, 146, 32expne0d 13504 . . . . . . . . . 10 (𝜑 → (𝐴𝑀) ≠ 0)
231165, 230div0d 11403 . . . . . . . . 9 (𝜑 → (0 / (𝐴𝑀)) = 0)
232231adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴𝑀)) = 0)
233229, 232eqtrd 2853 . . . . . . 7 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = 0)
234 fzfid 13329 . . . . . . 7 (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
235176, 178, 233, 234fsumss 15070 . . . . . 6 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
236133, 173, 2353eqtrd 2857 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
23786, 52syldan 591 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
23853fvmpt2 6771 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
23986, 237, 238syl2anc 584 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
240239adantlr 711 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
241 oveq1 7152 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑘) = (𝐴𝑘))
242241ad2antlr 723 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧𝑘) = (𝐴𝑘))
243240, 242oveq12d 7163 . . . . . . 7 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
244243sumeq2dv 15048 . . . . . 6 ((𝜑𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
245 fzfid 13329 . . . . . . 7 (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
246245, 129fsumcl 15078 . . . . . 6 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
2472, 244, 126, 246fvmptd 6767 . . . . 5 (𝜑 → (𝐹𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
24817, 16coeid2 24756 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
2495, 126, 248syl2anc 584 . . . . . . 7 (𝜑 → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
250249oveq1d 7160 . . . . . 6 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
25183adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
252182adantl 482 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
253251, 252ffvelrnd 6844 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
254126adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
255254, 252expcld 13498 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
256253, 255mulcld 10649 . . . . . . 7 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
257234, 165, 256, 230fsumdivc 15129 . . . . . 6 (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
258250, 257eqtrd 2853 . . . . 5 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
259236, 247, 2583eqtr4d 2863 . . . 4 (𝜑 → (𝐹𝐴) = ((𝐺𝐴) / (𝐴𝑀)))
260 elaa2lem.ga . . . . 5 (𝜑 → (𝐺𝐴) = 0)
261260oveq1d 7160 . . . 4 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
262259, 261, 2313eqtrd 2857 . . 3 (𝜑 → (𝐹𝐴) = 0)
263122, 262jca 512 . 2 (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0))
264 fveq2 6663 . . . . . 6 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
265264fveq1d 6665 . . . . 5 (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
266265neeq1d 3072 . . . 4 (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
267 fveq1 6662 . . . . 5 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
268267eqeq1d 2820 . . . 4 (𝑓 = 𝐹 → ((𝑓𝐴) = 0 ↔ (𝐹𝐴) = 0))
269266, 268anbi12d 630 . . 3 (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)))
270269rspcev 3620 . 2 ((𝐹 ∈ (Poly‘ℤ) ∧ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
27157, 263, 270syl2anc 584 1 (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  {crab 3139  cdif 3930  wss 3933  c0 4288  ifcif 4463   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  infcinf 8893  cc 10523  cr 10524  0cc0 10525   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  0cn0 11885  cz 11969  cuz 12231  ...cfz 12880  cexp 13417  Σcsu 15030  0𝑝c0p 24197  Polycply 24701  coeffccoe 24703  degcdgr 24704  𝔸caa 24830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-0p 24198  df-ply 24705  df-coe 24707  df-dgr 24708  df-aa 24831
This theorem is referenced by:  elaa2  42396
  Copyright terms: Public domain W3C validator