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Theorem elaa2lem 40953
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 40954. (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 11577 . . . . 5 ℤ ⊆ ℂ
43a1i 11 . . . 4 (𝜑 → ℤ ⊆ ℂ)
5 elaa2lem.g . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘ℤ))
6 dgrcl 24188 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
75, 6syl 17 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℕ0)
87nn0zd 11672 . . . . . . 7 (𝜑 → (deg‘𝐺) ∈ ℤ)
9 elaa2lem.m . . . . . . . . 9 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10 ssrab2 3828 . . . . . . . . . 10 {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11 nn0uz 11915 . . . . . . . . . . . . 13 0 = (ℤ‘0)
1210, 11sseqtri 3778 . . . . . . . . . . . 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 2937 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝐺 = 0𝑝)
16 eqid 2760 . . . . . . . . . . . . . . . . . 18 (deg‘𝐺) = (deg‘𝐺)
17 eqid 2760 . . . . . . . . . . . . . . . . . 18 (coeff‘𝐺) = (coeff‘𝐺)
1816, 17dgreq0 24220 . . . . . . . . . . . . . . . . 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 2939 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
227, 21jca 555 . . . . . . . . . . . . 13 (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
2423neeq1d 2991 . . . . . . . . . . . . . 14 (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2524elrab 3504 . . . . . . . . . . . . 13 ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
2622, 25sylibr 224 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
27 ne0i 4064 . . . . . . . . . . . 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 11965 . . . . . . . . . . 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 696 . . . . . . . . . 10 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
3110, 30sseldi 3742 . . . . . . . . 9 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
329, 31syl5eqel 2843 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3332nn0zd 11672 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
348, 33zsubcld 11679 . . . . . 6 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
359a1i 11 . . . . . . . 8 (𝜑𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
36 infssuzle 11964 . . . . . . . . 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 696 . . . . . . . 8 (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
3835, 37eqbrtrd 4826 . . . . . . 7 (𝜑𝑀 ≤ (deg‘𝐺))
397nn0red 11544 . . . . . . . 8 (𝜑 → (deg‘𝐺) ∈ ℝ)
4032nn0red 11544 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
4139, 40subge0d 10809 . . . . . . 7 (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
4238, 41mpbird 247 . . . . . 6 (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
4334, 42jca 555 . . . . 5 (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
44 elnn0z 11582 . . . . 5 (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
4543, 44sylibr 224 . . . 4 (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
46 id 22 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → 𝐺 ∈ (Poly‘ℤ))
47 0zd 11581 . . . . . . . . 9 (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
4817coef2 24186 . . . . . . . . 9 ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
4946, 47, 48syl2anc 696 . . . . . . . 8 (𝐺 ∈ (Poly‘ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
505, 49syl 17 . . . . . . 7 (𝜑 → (coeff‘𝐺):ℕ0⟶ℤ)
5150adantr 472 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (coeff‘𝐺):ℕ0⟶ℤ)
52 simpr 479 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
5332adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
5452, 53nn0addcld 11547 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
5551, 54ffvelrnd 6523 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
56 elaa2lem.i . . . . 5 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
5755, 56fmptd 6548 . . . 4 (𝜑𝐼:ℕ0⟶ℤ)
58 elplyr 24156 . . . 4 ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ0𝐼:ℕ0⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
594, 45, 57, 58syl3anc 1477 . . 3 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘))) ∈ (Poly‘ℤ))
602, 59eqeltrd 2839 . 2 (𝜑𝐹 ∈ (Poly‘ℤ))
61 simpr 479 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6261iftrued 4238 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
63 iffalse 4239 . . . . . . . . . . 11 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
6463adantl 473 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
65 simpr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
6639ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
6740ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
6866, 67resubcld 10650 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
69 nn0re 11493 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
7069ad2antlr 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
7168, 70ltnled 10376 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
7265, 71mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
7366, 67, 70ltsubaddd 10815 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
7472, 73mpbid 222 . . . . . . . . . . . . 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 764 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
7854adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ0)
7916, 17dgrlt 24221 . . . . . . . . . . . . 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 696 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
8176, 80mpbid 222 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
8281simprd 482 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
8364, 82eqtr4d 2797 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8462, 83pm2.61dan 867 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
8584mpteq2dva 4896 . . . . . . 7 (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
8650, 4fssd 6218 . . . . . . . . . 10 (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
8786adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
88 elfznn0 12626 . . . . . . . . . . 11 (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ0)
8988adantl 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ0)
9032adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ0)
9189, 90nn0addcld 11547 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
9287, 91ffvelrnd 6523 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
93 eqidd 2761 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
94 simpl 474 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝜑)
9556a1i 11 . . . . . . . . . . . . . . 15 (𝜑𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
9695, 55fvmpt2d 6455 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9794, 89, 96syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9897adantlr 753 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
9998oveq1d 6828 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
10093, 99sumeq12rdv 14637 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘)))
101100mpteq2dva 4896 . . . . . . . . 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 2794 . . . . . . . 8 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧𝑘))))
10360, 45, 92, 102coeeq2 24197 . . . . . . 7 (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
10485, 103, 953eqtr4d 2804 . . . . . 6 (𝜑 → (coeff‘𝐹) = 𝐼)
105104fveq1d 6354 . . . . 5 (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
106 oveq1 6820 . . . . . . . . 9 (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
107106adantl 473 . . . . . . . 8 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
1083, 33sseldi 3742 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
109108addid2d 10429 . . . . . . . . 9 (𝜑 → (0 + 𝑀) = 𝑀)
110109adantr 472 . . . . . . . 8 ((𝜑𝑘 = 0) → (0 + 𝑀) = 𝑀)
111107, 110eqtrd 2794 . . . . . . 7 ((𝜑𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
112111fveq2d 6356 . . . . . 6 ((𝜑𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
113 0nn0 11499 . . . . . . 7 0 ∈ ℕ0
114113a1i 11 . . . . . 6 (𝜑 → 0 ∈ ℕ0)
11550, 32ffvelrnd 6523 . . . . . 6 (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
11695, 112, 114, 115fvmptd 6450 . . . . 5 (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
117 eqidd 2761 . . . . 5 (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
118105, 116, 1173eqtrd 2798 . . . 4 (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
11935, 30eqeltrd 2839 . . . . . 6 (𝜑𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
120 fveq2 6352 . . . . . . . 8 (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
121120neeq1d 2991 . . . . . . 7 (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
122121elrab 3504 . . . . . 6 (𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
123119, 122sylib 208 . . . . 5 (𝜑 → (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
124123simprd 482 . . . 4 (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
125118, 124eqnetrd 2999 . . 3 (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
1265, 47syl 17 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
127 aasscn 24272 . . . . . . . . . . 11 𝔸 ⊆ ℂ
128 elaa2lem.a . . . . . . . . . . 11 (𝜑𝐴 ∈ 𝔸)
129127, 128sseldi 3742 . . . . . . . . . 10 (𝜑𝐴 ∈ ℂ)
13094, 129syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
131130, 89expcld 13202 . . . . . . . 8 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴𝑘) ∈ ℂ)
13292, 131mulcld 10252 . . . . . . 7 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
133 oveq1 6820 . . . . . . . . 9 (𝑘 = (𝑗𝑀) → (𝑘 + 𝑀) = ((𝑗𝑀) + 𝑀))
134133fveq2d 6356 . . . . . . . 8 (𝑘 = (𝑗𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)))
135 oveq2 6821 . . . . . . . 8 (𝑘 = (𝑗𝑀) → (𝐴𝑘) = (𝐴↑(𝑗𝑀)))
136134, 135oveq12d 6831 . . . . . . 7 (𝑘 = (𝑗𝑀) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))))
13733, 126, 34, 132, 136fsumshft 14711 . . . . . 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 3742 . . . . . . . . . 10 (𝜑 → (deg‘𝐺) ∈ ℂ)
139138, 108npcand 10588 . . . . . . . . 9 (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
140109, 139oveq12d 6831 . . . . . . . 8 (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
141140sumeq1d 14630 . . . . . . 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 12535 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
143142adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
1443, 143sseldi 3742 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
145108adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
146144, 145npcand 10588 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗𝑀) + 𝑀) = 𝑗)
147146fveq2d 6356 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
148147oveq1d 6828 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
149129adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
150 elaa2lem.an0 . . . . . . . . . . . . 13 (𝜑𝐴 ≠ 0)
151150adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
15233adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
153149, 151, 152, 143expsubd 13213 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗𝑀)) = ((𝐴𝑗) / (𝐴𝑀)))
154153oveq2d 6829 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
15586adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
156 0red 10233 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
15740adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
158143zred 11674 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
15932nn0ge0d 11546 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ≤ 𝑀)
160159adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
161 elfzle1 12537 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀𝑗)
162161adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀𝑗)
163156, 157, 158, 160, 162letrd 10386 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
164143, 163jca 555 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
165 elnn0z 11582 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
166164, 165sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
167155, 166ffvelrnd 6523 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
168149, 166expcld 13202 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
169129, 32expcld 13202 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑀) ∈ ℂ)
170169adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ∈ ℂ)
171149, 151, 152expne0d 13208 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴𝑀) ≠ 0)
172167, 168, 170, 171divassd 11028 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))))
173172eqcomd 2766 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴𝑗) / (𝐴𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
174154, 173eqtr2d 2795 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗𝑀))))
175148, 174eqtr4d 2797 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
176175sumeq2dv 14632 . . . . . . 7 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗𝑀) + 𝑀)) · (𝐴↑(𝑗𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
177141, 176eqtrd 2794 . . . . . 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 2849 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘0))
179 fzss1 12573 . . . . . . . 8 (𝑀 ∈ (ℤ‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
180178, 179syl 17 . . . . . . 7 (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
181167, 168mulcld 10252 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
182181, 170, 171divcld 10993 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) ∈ ℂ)
18333ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
1848ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
185 eldifi 3875 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
186 elfznn0 12626 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ0)
187186nn0zd 11672 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℤ)
188185, 187syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
189188ad2antlr 765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
190183, 184, 1893jca 1123 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ))
191 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
19240ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
193189zred 11674 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
194192, 193lenltd 10375 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
195191, 194mpbird 247 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀𝑗)
196 elfzle2 12538 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
197185, 196syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
198197ad2antlr 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
199190, 195, 198jca32 559 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
200 elfz2 12526 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀...(deg‘𝐺)) ↔ ((𝑀 ∈ ℤ ∧ (deg‘𝐺) ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝑀𝑗𝑗 ≤ (deg‘𝐺))))
201199, 200sylibr 224 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
202 eldifn 3876 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
203202ad2antlr 765 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
204201, 203condan 870 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
205204adantr 472 . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ0)
210 neqne 2940 . . . . . . . . . . . . . . . . . . 19 (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
211210adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
212209, 211jca 555 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
213 fveq2 6352 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
214213neeq1d 2991 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
215214elrab 3504 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
216212, 215sylibr 224 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
217216adantll 752 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
218 infssuzle 11964 . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
220206, 219eqbrtrd 4826 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀𝑗)
22140ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
222188zred 11674 . . . . . . . . . . . . . . 15 (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
223222ad2antlr 765 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
224221, 223lenltd 10375 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀𝑗 ↔ ¬ 𝑗 < 𝑀))
225220, 224mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
226205, 225condan 870 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
227226oveq1d 6828 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = (0 · (𝐴𝑗)))
228129adantr 472 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
229208adantl 473 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ0)
230228, 229expcld 13202 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴𝑗) ∈ ℂ)
231230mul02d 10426 . . . . . . . . . 10 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴𝑗)) = 0)
232227, 231eqtrd 2794 . . . . . . . . 9 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) = 0)
233232oveq1d 6828 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
234129, 150, 33expne0d 13208 . . . . . . . . . 10 (𝜑 → (𝐴𝑀) ≠ 0)
235169, 234div0d 10992 . . . . . . . . 9 (𝜑 → (0 / (𝐴𝑀)) = 0)
236235adantr 472 . . . . . . . 8 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴𝑀)) = 0)
237233, 236eqtrd 2794 . . . . . . 7 ((𝜑𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = 0)
238 fzfid 12966 . . . . . . 7 (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
239180, 182, 237, 238fsumss 14655 . . . . . 6 (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
240137, 177, 2393eqtrd 2798 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
24189, 55syldan 488 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
24256fvmpt2 6453 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
24389, 241, 242syl2anc 696 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
244243adantlr 753 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
245 oveq1 6820 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑧𝑘) = (𝐴𝑘))
246245ad2antlr 765 . . . . . . . 8 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧𝑘) = (𝐴𝑘))
247244, 246oveq12d 6831 . . . . . . 7 (((𝜑𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼𝑘) · (𝑧𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
248247sumeq2dv 14632 . . . . . 6 ((𝜑𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
249 fzfid 12966 . . . . . . 7 (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
250249, 132fsumcl 14663 . . . . . 6 (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)) ∈ ℂ)
2512, 248, 129, 250fvmptd 6450 . . . . 5 (𝜑 → (𝐹𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴𝑘)))
25217, 16coeid2 24194 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
2535, 129, 252syl2anc 696 . . . . . . 7 (𝜑 → (𝐺𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)))
254253oveq1d 6828 . . . . . 6 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
25586adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
256186adantl 473 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
257255, 256ffvelrnd 6523 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
258129adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
259258, 256expcld 13202 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (𝐴𝑗) ∈ ℂ)
260257, 259mulcld 10252 . . . . . . 7 ((𝜑𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) ∈ ℂ)
261238, 169, 260, 234fsumdivc 14717 . . . . . 6 (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
262254, 261eqtrd 2794 . . . . 5 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴𝑗)) / (𝐴𝑀)))
263240, 251, 2623eqtr4d 2804 . . . 4 (𝜑 → (𝐹𝐴) = ((𝐺𝐴) / (𝐴𝑀)))
264 elaa2lem.ga . . . . 5 (𝜑 → (𝐺𝐴) = 0)
265264oveq1d 6828 . . . 4 (𝜑 → ((𝐺𝐴) / (𝐴𝑀)) = (0 / (𝐴𝑀)))
266263, 265, 2353eqtrd 2798 . . 3 (𝜑 → (𝐹𝐴) = 0)
267125, 266jca 555 . 2 (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0))
268 fveq2 6352 . . . . . 6 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
269268fveq1d 6354 . . . . 5 (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
270269neeq1d 2991 . . . 4 (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
271 fveq1 6351 . . . . 5 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
272271eqeq1d 2762 . . . 4 (𝑓 = 𝐹 → ((𝑓𝐴) = 0 ↔ (𝐹𝐴) = 0))
273270, 272anbi12d 749 . . 3 (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)))
274273rspcev 3449 . 2 ((𝐹 ∈ (Poly‘ℤ) ∧ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
27560, 267, 274syl2anc 696 1 (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓𝐴) = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  {crab 3054  cdif 3712  wss 3715  c0 4058  ifcif 4230   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  infcinf 8512  cc 10126  cr 10127  0cc0 10128   + caddc 10131   · cmul 10133   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  0cn0 11484  cz 11569  cuz 11879  ...cfz 12519  cexp 13054  Σcsu 14615  0𝑝c0p 23635  Polycply 24139  coeffccoe 24141  degcdgr 24142  𝔸caa 24268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-0p 23636  df-ply 24143  df-coe 24145  df-dgr 24146  df-aa 24269
This theorem is referenced by:  elaa2  40954
  Copyright terms: Public domain W3C validator