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Theorem elaa2lem 39126
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 39127. (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 11262 . . . . 5 ℤ ⊆ ℂ
43a1i 11 . . . 4 (𝜑 → ℤ ⊆ ℂ)
5 elaa2lem.g . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘ℤ))
6 dgrcl 23793 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
75, 6syl 17 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℕ0)
87nn0zd 11356 . . . . . . 7 (𝜑 → (deg‘𝐺) ∈ ℤ)
9 elaa2lem.m . . . . . . . . 9 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10 ssrab2 3650 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11 nn0uz 11598 . . . . . . . . . . . . 13 0 = (ℤ‘0)
1210, 11sseqtri 3600 . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐺 = 0𝑝)
16 eqid 2610 . . . . . . . . . . . . . . . . . 18 (deg‘𝐺) = (deg‘𝐺)
17 eqid 2610 . . . . . . . . . . . . . . . . . 18 (coeff‘𝐺) = (coeff‘𝐺)
1816, 17dgreq0 23825 . . . . . . . . . . . . . . . . 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 313 . . . . . . . . . . . . . . 15 (𝜑 → ¬ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)
2120neqned 2789 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
227, 21jca 553 . . . . . . . . . . . . 13 (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
2423neeq1d 2841 . . . . . . . . . . . . . 14 (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2524elrab 3331 . . . . . . . . . . . . 13 ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2622, 25sylibr 223 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
27 ne0i 3880 . . . . . . . . . . . 12 ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
29 infssuzcl 11648 . . . . . . . . . . 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})
3013, 28, 29syl2anc 691 . . . . . . . . . 10 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
3110, 30sseldi 3566 . . . . . . . . 9 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
329, 31syl5eqel 2692 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3332nn0zd 11356 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
348, 33zsubcld 11363 . . . . . 6 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
359a1i 11 . . . . . . . 8 (𝜑𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
36 infssuzle 11647 . . . . . . . . 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))
3713, 26, 36syl2anc 691 . . . . . . . 8 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
3835, 37eqbrtrd 4605 . . . . . . 7 (𝜑𝑀 ≤ (deg‘𝐺))
397nn0red 11229 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℝ)
4032nn0red 11229 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
4139, 40subge0d 10496 . . . . . . 7 (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
4238, 41mpbird 246 . . . . . 6 (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
4334, 42jca 553 . . . . 5 (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
44 elnn0z 11267 . . . . 5 (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
4543, 44sylibr 223 . . . 4 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
46 id 22 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → 𝐺 ∈ (Poly‘ℤ))
47 0zd 11266 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
4817coef2 23791 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
4946, 47, 48syl2anc 691 . . . . . . . 8 (𝐺 ∈ (Poly‘ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
505, 49syl 17 . . . . . . 7 (𝜑 → (coeff‘𝐺):ℕ0⟶ℤ)
5150adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (coeff‘𝐺):ℕ0⟶ℤ)
52 simpr 476 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
5332adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
5452, 53nn0addcld 11232 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5551, 54ffvelrnd 6268 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
56 elaa2lem.i . . . . 5 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
5755, 56fmptd 6292 . . . 4 (𝜑𝐼:ℕ0⟶ℤ)
58 elplyr 23761 . . . 4 ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ0𝐼:ℕ0⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
594, 45, 57, 58syl3anc 1318 . . 3 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
602, 59eqeltrd 2688 . 2 (𝜑𝐹 ∈ (Poly‘ℤ))
61 simpr 476 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6261iftrued 4044 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
63 iffalse 4045 . . . . . . . . . . 11 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
6463adantl 481 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
65 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6639ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
6740ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
6866, 67resubcld 10337 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
69 nn0re 11178 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
7069ad2antlr 759 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
7168, 70ltnled 10063 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
7265, 71mpbird 246 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
7366, 67, 70ltsubaddd 10502 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
7472, 73mpbid 221 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) < (𝑘 + 𝑀))
75 olc 398 . . . . . . . . . . . . 13 ((deg‘𝐺) < (𝑘 + 𝑀) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
7674, 75syl 17 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
775ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
7854adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ0)
7916, 17dgrlt 23826 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℤ) ∧ (𝑘 + 𝑀) ∈ ℕ0) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
8077, 78, 79syl2anc 691 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
8176, 80mpbid 221 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
8281simprd 478 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
8364, 82eqtr4d 2647 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8462, 83pm2.61dan 828 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8584mpteq2dva 4672 . . . . . . 7 (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
8650, 4fssd 5970 . . . . . . . . . 10 (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
8786adantr 480 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
88 elfznn0 12302 . . . . . . . . . . 11 (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ0)
8988adantl 481 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ0)
9032adantr 480 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ0)
9189, 90nn0addcld 11232 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
9287, 91ffvelrnd 6268 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
93 eqidd 2611 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
94 simpl 472 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝜑)
9556a1i 11 . . . . . . . . . . . . . . 15 (𝜑𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
9695, 55fvmpt2d 6202 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9794, 89, 96syl2anc 691 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9897adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9998oveq1d 6564 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
10093, 99sumeq12rdv 14285 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
101100mpteq2dva 4672 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
1022, 101eqtrd 2644 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
10360, 45, 92, 102coeeq2 23802 . . . . . . 7 (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
10485, 103, 953eqtr4d 2654 . . . . . 6 (𝜑 → (coeff‘𝐹) = 𝐼)
105104fveq1d 6105 . . . . 5 (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
106 oveq1 6556 . . . . . . . . 9 (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
107106adantl 481 . . . . . . . 8 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
1083, 33sseldi 3566 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
109108addid2d 10116 . . . . . . . . 9 (𝜑 → (0 + 𝑀) = 𝑀)
110109adantr 480 . . . . . . . 8 ((𝜑𝑘 = 0) → (0 + 𝑀) = 𝑀)
111107, 110eqtrd 2644 . . . . . . 7 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
112111fveq2d 6107 . . . . . 6 ((𝜑𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
113 0nn0 11184 . . . . . . 7 0 ∈ ℕ0
114113a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℕ0)
11550, 32ffvelrnd 6268 . . . . . 6 (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
11695, 112, 114, 115fvmptd 6197 . . . . 5 (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
117 eqidd 2611 . . . . 5 (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
118105, 116, 1173eqtrd 2648 . . . 4 (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
11935, 30eqeltrd 2688 . . . . . 6 (𝜑𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
120 fveq2 6103 . . . . . . . 8 (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
121120neeq1d 2841 . . . . . . 7 (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
122121elrab 3331 . . . . . 6 (𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
123119, 122sylib 207 . . . . 5 (𝜑 → (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
124123simprd 478 . . . 4 (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
125118, 124eqnetrd 2849 . . 3 (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
1265, 47syl 17 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
127 aasscn 23877 . . . . . . . . . . 11 𝔸 ⊆ ℂ
128 elaa2lem.a . . . . . . . . . . 11 (𝜑𝐴 ∈ 𝔸)
129127, 128sseldi 3566 . . . . . . . . . 10 (𝜑𝐴 ∈ ℂ)
13094, 129syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
131130, 89expcld 12870 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴𝑘) ∈ ℂ)
13292, 131mulcld 9939 . . . . . . 7 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
133 oveq1 6556 . . . . . . . . 9 (𝑘 = (𝑗𝑀) → (𝑘 + 𝑀) = ((𝑗𝑀) + 𝑀))
134133fveq2d 6107 . . . . . . . 8 (𝑘 = (𝑗𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)))
135 oveq2 6557 . . . . . . . 8 (𝑘 = (𝑗𝑀) → (𝐴𝑘) = (𝐴↑(𝑗𝑀)))
136134, 135oveq12d 6567 . . . . . . 7 (𝑘 = (𝑗𝑀) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
13733, 126, 34, 132, 136fsumshft 14354 . . . . . 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))))
1383, 8sseldi 3566 . . . . . . . . . 10 (𝜑 → (deg‘𝐺) ∈ ℂ)
139138, 108npcand 10275 . . . . . . . . 9 (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
140109, 139oveq12d 6567 . . . . . . . 8 (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
141140sumeq1d 14279 . . . . . . 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))))
142 elfzelz 12213 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
143142adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
1443, 143sseldi 3566 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
145108adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
146144, 145npcand 10275 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗𝑀) + 𝑀) = 𝑗)
147146fveq2d 6107 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
148147oveq1d 6564 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
149129adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
150 elaa2lem.an0 . . . . . . . . . . . . 13 (𝜑𝐴 ≠ 0)
151150adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
15233adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
153149, 151, 152, 143expsubd 12881 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗𝑀)) = ((𝐴𝑗) / (𝐴𝑀)))
154153oveq2d 6565 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
15586adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
156 0red 9920 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
15740adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
158143zred 11358 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
15932nn0ge0d 11231 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ≤ 𝑀)
160159adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
161 elfzle1 12215 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀𝑗)
162161adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀𝑗)
163156, 157, 158, 160, 162letrd 10073 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
164143, 163jca 553 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
165 elnn0z 11267 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
166164, 165sylibr 223 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
167155, 166ffvelrnd 6268 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
168149, 166expcld 12870 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
169129, 32expcld 12870 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑀) ∈ ℂ)
170169adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ∈ ℂ)
171149, 151, 152expne0d 12876 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ≠ 0)
172167, 168, 170, 171divassd 10715 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
173172eqcomd 2616 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
174154, 173eqtr2d 2645 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
175148, 174eqtr4d 2647 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
176175sumeq2dv 14281 . . . . . . 7 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
177141, 176eqtrd 2644 . . . . . 6 (𝜑 → Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
17832, 11syl6eleq 2698 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘0))
179 fzss1 12251 . . . . . . . 8 (𝑀 ∈ (ℤ‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
180178, 179syl 17 . . . . . . 7 (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
181167, 168mulcld 9939 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
182181, 170, 171divcld 10680 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) ∈ ℂ)
18333ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
1848ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
185 eldifi 3694 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
186 elfznn0 12302 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ0)
187186nn0zd 11356 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℤ)
188185, 187syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
189188ad2antlr 759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
190183, 184, 1893jca 1235 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ))
191 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
19240ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
193189zred 11358 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
194192, 193lenltd 10062 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
195191, 194mpbird 246 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀𝑗)
196 elfzle2 12216 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
197185, 196syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
198197ad2antlr 759 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
199190, 195, 198jca32 556 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
200 elfz2 12204 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀...(deg‘𝐺)) ↔ ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
201199, 200sylibr 223 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
202 eldifn 3695 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
203202ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
204201, 203condan 831 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
205204adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 < 𝑀)
2069a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
20712a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ‘0))
208185, 186syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
209208adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ0)
210 neqne 2790 . . . . . . . . . . . . . . . . . . 19 (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
211210adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
212209, 211jca 553 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
213 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
214213neeq1d 2841 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
215214elrab 3331 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
216212, 215sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
217216adantll 746 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
218 infssuzle 11647 . . . . . . . . . . . . . . 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)
219207, 217, 218syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
220206, 219eqbrtrd 4605 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀𝑗)
22140ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
222188zred 11358 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
223222ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
224221, 223lenltd 10062 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
225220, 224mpbid 221 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
226205, 225condan 831 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
227226oveq1d 6564 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = (0 · (𝐴𝑗)))
228129adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
229208adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ0)
230228, 229expcld 12870 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴𝑗) ∈ ℂ)
231230mul02d 10113 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴𝑗)) = 0)
232227, 231eqtrd 2644 . . . . . . . . 9 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = 0)
233232oveq1d 6564 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
234129, 150, 33expne0d 12876 . . . . . . . . . 10 (𝜑 → (𝐴𝑀) ≠ 0)
235169, 234div0d 10679 . . . . . . . . 9 (𝜑 → (0 / (𝐴𝑀)) = 0)
236235adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴𝑀)) = 0)
237233, 236eqtrd 2644 . . . . . . 7 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = 0)
238 fzfid 12634 . . . . . . 7 (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
239180, 182, 237, 238fsumss 14303 . . . . . 6 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
240137, 177, 2393eqtrd 2648 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
24189, 55syldan 486 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
24256fvmpt2 6200 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
24389, 241, 242syl2anc 691 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
244243adantlr 747 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
245 oveq1 6556 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑘) = (𝐴𝑘))
246245ad2antlr 759 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧𝑘) = (𝐴𝑘))
247244, 246oveq12d 6567 . . . . . . 7 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
248247sumeq2dv 14281 . . . . . 6 ((𝜑𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
249 fzfid 12634 . . . . . . 7 (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
250249, 132fsumcl 14311 . . . . . 6 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
2512, 248, 129, 250fvmptd 6197 . . . . 5 (𝜑 → (𝐹𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
25217, 16coeid2 23799 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
2535, 129, 252syl2anc 691 . . . . . . 7 (𝜑 → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
254253oveq1d 6564 . . . . . 6 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
25586adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
256186adantl 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
257255, 256ffvelrnd 6268 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
258129adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
259258, 256expcld 12870 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
260257, 259mulcld 9939 . . . . . . 7 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
261238, 169, 260, 234fsumdivc 14360 . . . . . 6 (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
262254, 261eqtrd 2644 . . . . 5 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
263240, 251, 2623eqtr4d 2654 . . . 4 (𝜑 → (𝐹𝐴) = ((𝐺𝐴) / (𝐴𝑀)))
264 elaa2lem.ga . . . . 5 (𝜑 → (𝐺𝐴) = 0)
265264oveq1d 6564 . . . 4 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
266263, 265, 2353eqtrd 2648 . . 3 (𝜑 → (𝐹𝐴) = 0)
267125, 266jca 553 . 2 (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0))
268 fveq2 6103 . . . . . 6 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
269268fveq1d 6105 . . . . 5 (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
270269neeq1d 2841 . . . 4 (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
271 fveq1 6102 . . . . 5 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
272271eqeq1d 2612 . . . 4 (𝑓 = 𝐹 → ((𝑓𝐴) = 0 ↔ (𝐹𝐴) = 0))
273270, 272anbi12d 743 . . 3 (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)))
274273rspcev 3282 . 2 ((𝐹 ∈ (Poly‘ℤ) ∧ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
27560, 267, 274syl2anc 691 1 (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897  {crab 2900  cdif 3537  wss 3540  c0 3874  ifcif 4036   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  infcinf 8230  cc 9813  cr 9814  0cc0 9815   + caddc 9818   · cmul 9820   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  0cn0 11169  cz 11254  cuz 11563  ...cfz 12197  cexp 12722  Σcsu 14264  0𝑝c0p 23242  Polycply 23744  coeffccoe 23746  degcdgr 23747  𝔸caa 23873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751  df-aa 23874
This theorem is referenced by:  elaa2  39127
  Copyright terms: Public domain W3C validator