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Theorem elaa2lem 46189
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 46190. (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 12619 . . . . 5 ℤ ⊆ ℂ
43a1i 11 . . . 4 (𝜑 → ℤ ⊆ ℂ)
5 elaa2lem.g . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘ℤ))
6 dgrcl 26287 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
75, 6syl 17 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℕ0)
87nn0zd 12637 . . . . . . 7 (𝜑 → (deg‘𝐺) ∈ ℤ)
9 elaa2lem.m . . . . . . . . 9 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10 ssrab2 4090 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11 nn0uz 12918 . . . . . . . . . . . . 13 0 = (ℤ‘0)
1210, 11sseqtri 4032 . . . . . . . . . . . 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 2943 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐺 = 0𝑝)
16 eqid 2735 . . . . . . . . . . . . . . . . . 18 (deg‘𝐺) = (deg‘𝐺)
17 eqid 2735 . . . . . . . . . . . . . . . . . 18 (coeff‘𝐺) = (coeff‘𝐺)
1816, 17dgreq0 26320 . . . . . . . . . . . . . . . . 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 324 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)
2120neqned 2945 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
227, 21jca 511 . . . . . . . . . . . . 13 (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
2423neeq1d 2998 . . . . . . . . . . . . . 14 (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2524elrab 3695 . . . . . . . . . . . . 13 ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2622, 25sylibr 234 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
2726ne0d 4348 . . . . . . . . . . 11 (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
28 infssuzcl 12972 . . . . . . . . . . 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, 29sselid 3993 . . . . . . . . 9 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
319, 30eqeltrid 2843 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3231nn0zd 12637 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
338, 32zsubcld 12725 . . . . . 6 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
349a1i 11 . . . . . . . 8 (𝜑𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
35 infssuzle 12971 . . . . . . . . 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 5170 . . . . . . 7 (𝜑𝑀 ≤ (deg‘𝐺))
387nn0red 12586 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℝ)
3931nn0red 12586 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
4038, 39subge0d 11851 . . . . . . 7 (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
4137, 40mpbird 257 . . . . . 6 (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
4233, 41jca 511 . . . . 5 (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
43 elnn0z 12624 . . . . 5 (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
4442, 43sylibr 234 . . . 4 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
45 0zd 12623 . . . . . . . 8 (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
4617coef2 26285 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
475, 45, 46syl2anc2 585 . . . . . . 7 (𝜑 → (coeff‘𝐺):ℕ0⟶ℤ)
4847adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (coeff‘𝐺):ℕ0⟶ℤ)
49 simpr 484 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
5031adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
5149, 50nn0addcld 12589 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5248, 51ffvelcdmd 7105 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
53 elaa2lem.i . . . . 5 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
5452, 53fmptd 7134 . . . 4 (𝜑𝐼:ℕ0⟶ℤ)
55 elplyr 26255 . . . 4 ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ0𝐼:ℕ0⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
564, 44, 54, 55syl3anc 1370 . . 3 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
572, 56eqeltrd 2839 . 2 (𝜑𝐹 ∈ (Poly‘ℤ))
58 simpr 484 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
5958iftrued 4539 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
60 iffalse 4540 . . . . . . . . . . 11 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
6160adantl 481 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
62 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6338ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
6439ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
6563, 64resubcld 11689 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
66 nn0re 12533 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
6766ad2antlr 727 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
6865, 67ltnled 11406 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
6962, 68mpbird 257 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
7063, 64, 67ltsubaddd 11857 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
7169, 70mpbid 232 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) < (𝑘 + 𝑀))
72 olc 868 . . . . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
7551adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ0)
7616, 17dgrlt 26321 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
7978simprd 495 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
8061, 79eqtr4d 2778 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8159, 80pm2.61dan 813 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8281mpteq2dva 5248 . . . . . . 7 (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
8347, 4fssd 6754 . . . . . . . . . 10 (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
8483adantr 480 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
85 elfznn0 13657 . . . . . . . . . . 11 (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ0)
8685adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ0)
8731adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ0)
8886, 87nn0addcld 12589 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
8984, 88ffvelcdmd 7105 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
90 eqidd 2736 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
91 simpl 482 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝜑)
9253a1i 11 . . . . . . . . . . . . . . 15 (𝜑𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
9392, 52fvmpt2d 7029 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9491, 86, 93syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9594adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9695oveq1d 7446 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
9790, 96sumeq12rdv 15740 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
9897mpteq2dva 5248 . . . . . . . . 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 2775 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
10057, 44, 89, 99coeeq2 26296 . . . . . . 7 (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
10182, 100, 923eqtr4d 2785 . . . . . 6 (𝜑 → (coeff‘𝐹) = 𝐼)
102101fveq1d 6909 . . . . 5 (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
103 oveq1 7438 . . . . . . . . 9 (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
104103adantl 481 . . . . . . . 8 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
1053, 32sselid 3993 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
106105addlidd 11460 . . . . . . . . 9 (𝜑 → (0 + 𝑀) = 𝑀)
107106adantr 480 . . . . . . . 8 ((𝜑𝑘 = 0) → (0 + 𝑀) = 𝑀)
108104, 107eqtrd 2775 . . . . . . 7 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
109108fveq2d 6911 . . . . . 6 ((𝜑𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
110 0nn0 12539 . . . . . . 7 0 ∈ ℕ0
111110a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℕ0)
11247, 31ffvelcdmd 7105 . . . . . 6 (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
11392, 109, 111, 112fvmptd 7023 . . . . 5 (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
114 eqidd 2736 . . . . 5 (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
115102, 113, 1143eqtrd 2779 . . . 4 (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
11634, 29eqeltrd 2839 . . . . . 6 (𝜑𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
117 fveq2 6907 . . . . . . . 8 (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
118117neeq1d 2998 . . . . . . 7 (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
119118elrab 3695 . . . . . 6 (𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
120116, 119sylib 218 . . . . 5 (𝜑 → (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
121120simprd 495 . . . 4 (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
122115, 121eqnetrd 3006 . . 3 (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
1235, 45syl 17 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
124 aasscn 26375 . . . . . . . . . . 11 𝔸 ⊆ ℂ
125 elaa2lem.a . . . . . . . . . . 11 (𝜑𝐴 ∈ 𝔸)
126124, 125sselid 3993 . . . . . . . . . 10 (𝜑𝐴 ∈ ℂ)
12791, 126syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
128127, 86expcld 14183 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴𝑘) ∈ ℂ)
12989, 128mulcld 11279 . . . . . . 7 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
130 fvoveq1 7454 . . . . . . . 8 (𝑘 = (𝑗𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)))
131 oveq2 7439 . . . . . . . 8 (𝑘 = (𝑗𝑀) → (𝐴𝑘) = (𝐴↑(𝑗𝑀)))
132130, 131oveq12d 7449 . . . . . . 7 (𝑘 = (𝑗𝑀) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
13332, 123, 33, 129, 132fsumshft 15813 . . . . . 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, 8sselid 3993 . . . . . . . . . 10 (𝜑 → (deg‘𝐺) ∈ ℂ)
135134, 105npcand 11622 . . . . . . . . 9 (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
136106, 135oveq12d 7449 . . . . . . . 8 (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
137136sumeq1d 15733 . . . . . . 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 13561 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
139138adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
1403, 139sselid 3993 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
141105adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
142140, 141npcand 11622 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗𝑀) + 𝑀) = 𝑗)
143142fveq2d 6911 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
144143oveq1d 7446 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
145126adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
146 elaa2lem.an0 . . . . . . . . . . . . 13 (𝜑𝐴 ≠ 0)
147146adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
14832adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
149145, 147, 148, 139expsubd 14194 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗𝑀)) = ((𝐴𝑗) / (𝐴𝑀)))
150149oveq2d 7447 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
15183adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
152 0red 11262 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
15339adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
154139zred 12720 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
15531nn0ge0d 12588 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ≤ 𝑀)
156155adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
157 elfzle1 13564 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀𝑗)
158157adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀𝑗)
159152, 153, 154, 156, 158letrd 11416 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
160139, 159jca 511 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
161 elnn0z 12624 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
162160, 161sylibr 234 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
163151, 162ffvelcdmd 7105 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
164145, 162expcld 14183 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
165126, 31expcld 14183 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑀) ∈ ℂ)
166165adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ∈ ℂ)
167145, 147, 148expne0d 14189 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ≠ 0)
168163, 164, 166, 167divassd 12076 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
169168eqcomd 2741 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
170150, 169eqtr2d 2776 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
171144, 170eqtr4d 2778 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
172171sumeq2dv 15735 . . . . . . 7 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
173137, 172eqtrd 2775 . . . . . 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 2849 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘0))
175 fzss1 13600 . . . . . . . 8 (𝑀 ∈ (ℤ‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
176174, 175syl 17 . . . . . . 7 (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
177163, 164mulcld 11279 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
178177, 166, 167divcld 12041 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) ∈ ℂ)
17932ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
1808ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
181 eldifi 4141 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
182181elfzelzd 13562 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
183182ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
184 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
18539ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
186183zred 12720 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
187185, 186lenltd 11405 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
188184, 187mpbird 257 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀𝑗)
189 elfzle2 13565 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
190181, 189syl 17 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
191190ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
192179, 180, 183, 188, 191elfzd 13552 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
193 eldifn 4142 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
194193ad2antlr 727 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
195192, 194condan 818 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
196195adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 < 𝑀)
1979a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
19812a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0))
199 elfznn0 13657 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ0)
200181, 199syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
201200adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ0)
202 neqne 2946 . . . . . . . . . . . . . . . . . . 19 (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
203202adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
204201, 203jca 511 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
205 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
206205neeq1d 2998 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
207206elrab 3695 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
208204, 207sylibr 234 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
209208adantll 714 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
210 infssuzle 12971 . . . . . . . . . . . . . . 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)
211198, 209, 210syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
212197, 211eqbrtrd 5170 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀𝑗)
21339ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
214182zred 12720 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
215214ad2antlr 727 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
216213, 215lenltd 11405 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
217212, 216mpbid 232 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
218196, 217condan 818 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
219218oveq1d 7446 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = (0 · (𝐴𝑗)))
220126adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
221200adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ0)
222220, 221expcld 14183 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴𝑗) ∈ ℂ)
223222mul02d 11457 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴𝑗)) = 0)
224219, 223eqtrd 2775 . . . . . . . . 9 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = 0)
225224oveq1d 7446 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
226126, 146, 32expne0d 14189 . . . . . . . . . 10 (𝜑 → (𝐴𝑀) ≠ 0)
227165, 226div0d 12040 . . . . . . . . 9 (𝜑 → (0 / (𝐴𝑀)) = 0)
228227adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴𝑀)) = 0)
229225, 228eqtrd 2775 . . . . . . 7 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = 0)
230 fzfid 14011 . . . . . . 7 (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
231176, 178, 229, 230fsumss 15758 . . . . . 6 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
232133, 173, 2313eqtrd 2779 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
23386, 52syldan 591 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
23453fvmpt2 7027 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
23586, 233, 234syl2anc 584 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
236235adantlr 715 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
237 oveq1 7438 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑘) = (𝐴𝑘))
238237ad2antlr 727 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧𝑘) = (𝐴𝑘))
239236, 238oveq12d 7449 . . . . . . 7 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
240239sumeq2dv 15735 . . . . . 6 ((𝜑𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
241 fzfid 14011 . . . . . . 7 (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
242241, 129fsumcl 15766 . . . . . 6 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
2432, 240, 126, 242fvmptd 7023 . . . . 5 (𝜑 → (𝐹𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
24417, 16coeid2 26293 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
2455, 126, 244syl2anc 584 . . . . . . 7 (𝜑 → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
246245oveq1d 7446 . . . . . 6 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
24783adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
248199adantl 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
249247, 248ffvelcdmd 7105 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
250126adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
251250, 248expcld 14183 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
252249, 251mulcld 11279 . . . . . . 7 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
253230, 165, 252, 226fsumdivc 15819 . . . . . 6 (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
254246, 253eqtrd 2775 . . . . 5 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
255232, 243, 2543eqtr4d 2785 . . . 4 (𝜑 → (𝐹𝐴) = ((𝐺𝐴) / (𝐴𝑀)))
256 elaa2lem.ga . . . . 5 (𝜑 → (𝐺𝐴) = 0)
257256oveq1d 7446 . . . 4 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
258255, 257, 2273eqtrd 2779 . . 3 (𝜑 → (𝐹𝐴) = 0)
259122, 258jca 511 . 2 (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0))
260 fveq2 6907 . . . . . 6 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
261260fveq1d 6909 . . . . 5 (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
262261neeq1d 2998 . . . 4 (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
263 fveq1 6906 . . . . 5 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
264263eqeq1d 2737 . . . 4 (𝑓 = 𝐹 → ((𝑓𝐴) = 0 ↔ (𝐹𝐴) = 0))
265262, 264anbi12d 632 . . 3 (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)))
266265rspcev 3622 . 2 ((𝐹 ∈ (Poly‘ℤ) ∧ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
26757, 259, 266syl2anc 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  cdif 3960  wss 3963  c0 4339  ifcif 4531   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  infcinf 9479  cc 11151  cr 11152  0cc0 11153   + caddc 11156   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  0cn0 12524  cz 12611  cuz 12876  ...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:  elaa2  46190
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