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Theorem elaa2lem 46248
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 46249. (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 12621 . . . . 5 ℤ ⊆ ℂ
43a1i 11 . . . 4 (𝜑 → ℤ ⊆ ℂ)
5 elaa2lem.g . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘ℤ))
6 dgrcl 26272 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
75, 6syl 17 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℕ0)
87nn0zd 12639 . . . . . . 7 (𝜑 → (deg‘𝐺) ∈ ℤ)
9 elaa2lem.m . . . . . . . . 9 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10 ssrab2 4080 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11 nn0uz 12920 . . . . . . . . . . . . 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 2945 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐺 = 0𝑝)
16 eqid 2737 . . . . . . . . . . . . . . . . . 18 (deg‘𝐺) = (deg‘𝐺)
17 eqid 2737 . . . . . . . . . . . . . . . . . 18 (coeff‘𝐺) = (coeff‘𝐺)
1816, 17dgreq0 26305 . . . . . . . . . . . . . . . . 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 2947 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
227, 21jca 511 . . . . . . . . . . . . 13 (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
2423neeq1d 3000 . . . . . . . . . . . . . 14 (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2524elrab 3692 . . . . . . . . . . . . 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 4342 . . . . . . . . . . 11 (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
28 infssuzcl 12974 . . . . . . . . . . 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 3981 . . . . . . . . 9 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
319, 30eqeltrid 2845 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3231nn0zd 12639 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
338, 32zsubcld 12727 . . . . . 6 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
349a1i 11 . . . . . . . 8 (𝜑𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
35 infssuzle 12973 . . . . . . . . 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 5165 . . . . . . 7 (𝜑𝑀 ≤ (deg‘𝐺))
387nn0red 12588 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℝ)
3931nn0red 12588 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
4038, 39subge0d 11853 . . . . . . 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 12626 . . . . 5 (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
4442, 43sylibr 234 . . . 4 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
45 0zd 12625 . . . . . . . 8 (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
4617coef2 26270 . . . . . . . 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 12591 . . . . . 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 26240 . . . 4 ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ0𝐼:ℕ0⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
564, 44, 54, 55syl3anc 1373 . . 3 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
572, 56eqeltrd 2841 . 2 (𝜑𝐹 ∈ (Poly‘ℤ))
58 simpr 484 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
5958iftrued 4533 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
60 iffalse 4534 . . . . . . . . . . 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 11691 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
66 nn0re 12535 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
6766ad2antlr 727 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
6865, 67ltnled 11408 . . . . . . . . . . . . . . 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 11859 . . . . . . . . . . . . . 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 869 . . . . . . . . . . . . 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 26306 . . . . . . . . . . . . 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 2780 . . . . . . . . 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 5242 . . . . . . 7 (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
8347, 4fssd 6753 . . . . . . . . . 10 (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
8483adantr 480 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
85 elfznn0 13660 . . . . . . . . . . 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 12591 . . . . . . . . 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 2738 . . . . . . . . . . 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 15743 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
9897mpteq2dva 5242 . . . . . . . . 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 2777 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
10057, 44, 89, 99coeeq2 26281 . . . . . . 7 (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
10182, 100, 923eqtr4d 2787 . . . . . 6 (𝜑 → (coeff‘𝐹) = 𝐼)
102101fveq1d 6908 . . . . 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 3981 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
106105addlidd 11462 . . . . . . . . 9 (𝜑 → (0 + 𝑀) = 𝑀)
107106adantr 480 . . . . . . . 8 ((𝜑𝑘 = 0) → (0 + 𝑀) = 𝑀)
108104, 107eqtrd 2777 . . . . . . 7 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
109108fveq2d 6910 . . . . . 6 ((𝜑𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
110 0nn0 12541 . . . . . . 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 2738 . . . . 5 (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
115102, 113, 1143eqtrd 2781 . . . 4 (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
11634, 29eqeltrd 2841 . . . . . 6 (𝜑𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
117 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
118117neeq1d 3000 . . . . . . 7 (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
119118elrab 3692 . . . . . 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 3008 . . 3 (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
1235, 45syl 17 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
124 aasscn 26360 . . . . . . . . . . 11 𝔸 ⊆ ℂ
125 elaa2lem.a . . . . . . . . . . 11 (𝜑𝐴 ∈ 𝔸)
126124, 125sselid 3981 . . . . . . . . . 10 (𝜑𝐴 ∈ ℂ)
12791, 126syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
128127, 86expcld 14186 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴𝑘) ∈ ℂ)
12989, 128mulcld 11281 . . . . . . 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 15816 . . . . . 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 3981 . . . . . . . . . 10 (𝜑 → (deg‘𝐺) ∈ ℂ)
135134, 105npcand 11624 . . . . . . . . 9 (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
136106, 135oveq12d 7449 . . . . . . . 8 (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
137136sumeq1d 15736 . . . . . . 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 13564 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
139138adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
1403, 139sselid 3981 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
141105adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
142140, 141npcand 11624 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗𝑀) + 𝑀) = 𝑗)
143142fveq2d 6910 . . . . . . . . . 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 14197 . . . . . . . . . . 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 11264 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
15339adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
154139zred 12722 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
15531nn0ge0d 12590 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ≤ 𝑀)
156155adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
157 elfzle1 13567 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀𝑗)
158157adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀𝑗)
159152, 153, 154, 156, 158letrd 11418 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
160139, 159jca 511 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
161 elnn0z 12626 . . . . . . . . . . . . . 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 14186 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
165126, 31expcld 14186 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑀) ∈ ℂ)
166165adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ∈ ℂ)
167145, 147, 148expne0d 14192 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ≠ 0)
168163, 164, 166, 167divassd 12078 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
169168eqcomd 2743 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
170150, 169eqtr2d 2778 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
171144, 170eqtr4d 2780 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
172171sumeq2dv 15738 . . . . . . 7 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
173137, 172eqtrd 2777 . . . . . 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 2851 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘0))
175 fzss1 13603 . . . . . . . 8 (𝑀 ∈ (ℤ‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
176174, 175syl 17 . . . . . . 7 (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
177163, 164mulcld 11281 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
178177, 166, 167divcld 12043 . . . . . . 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 4131 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
182181elfzelzd 13565 . . . . . . . . . . . . . . . 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 12722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
187185, 186lenltd 11407 . . . . . . . . . . . . . . . 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 13568 . . . . . . . . . . . . . . . . 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 13555 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
193 eldifn 4132 . . . . . . . . . . . . . . 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 13660 . . . . . . . . . . . . . . . . . . . 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 2948 . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
206205neeq1d 3000 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
207206elrab 3692 . . . . . . . . . . . . . . . . 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 12973 . . . . . . . . . . . . . . 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 5165 . . . . . . . . . . . . 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 12722 . . . . . . . . . . . . . . 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 11407 . . . . . . . . . . . . 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 14186 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴𝑗) ∈ ℂ)
223222mul02d 11459 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴𝑗)) = 0)
224219, 223eqtrd 2777 . . . . . . . . 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 14192 . . . . . . . . . 10 (𝜑 → (𝐴𝑀) ≠ 0)
227165, 226div0d 12042 . . . . . . . . 9 (𝜑 → (0 / (𝐴𝑀)) = 0)
228227adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴𝑀)) = 0)
229225, 228eqtrd 2777 . . . . . . 7 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = 0)
230 fzfid 14014 . . . . . . 7 (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
231176, 178, 229, 230fsumss 15761 . . . . . 6 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
232133, 173, 2313eqtrd 2781 . . . . 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 15738 . . . . . 6 ((𝜑𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
241 fzfid 14014 . . . . . . 7 (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
242241, 129fsumcl 15769 . . . . . 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 26278 . . . . . . . 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 14186 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
252249, 251mulcld 11281 . . . . . . 7 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
253230, 165, 252, 226fsumdivc 15822 . . . . . 6 (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
254246, 253eqtrd 2777 . . . . 5 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
255232, 243, 2543eqtr4d 2787 . . . 4 (𝜑 → (𝐹𝐴) = ((𝐺𝐴) / (𝐴𝑀)))
256 elaa2lem.ga . . . . 5 (𝜑 → (𝐺𝐴) = 0)
257256oveq1d 7446 . . . 4 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
258255, 257, 2273eqtrd 2781 . . 3 (𝜑 → (𝐹𝐴) = 0)
259122, 258jca 511 . 2 (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0))
260 fveq2 6906 . . . . . 6 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
261260fveq1d 6908 . . . . 5 (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
262261neeq1d 3000 . . . 4 (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
263 fveq1 6905 . . . . 5 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
264263eqeq1d 2739 . . . 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 848   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  cdif 3948  wss 3951  c0 4333  ifcif 4525   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  infcinf 9481  cc 11153  cr 11154  0cc0 11155   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  cexp 14102  Σcsu 15722  0𝑝c0p 25704  Polycply 26223  coeffccoe 26225  degcdgr 26226  𝔸caa 26356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-coe 26229  df-dgr 26230  df-aa 26357
This theorem is referenced by:  elaa2  46249
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