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Theorem aannenlem2 24929
 Description: Lemma for aannen 24931. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypothesis
Ref Expression
aannenlem.a 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
Assertion
Ref Expression
aannenlem2 𝔸 = ran 𝐻
Distinct variable group:   𝑎,𝑏,𝑐,𝑑,𝑒
Allowed substitution hints:   𝐻(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aannenlem2
Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6658 . . . . . . . . . . 11 (𝑏 = 𝑔 → ((𝑐𝑏) = 0 ↔ (𝑐𝑔) = 0))
21rexbidv 3259 . . . . . . . . . 10 (𝑏 = 𝑔 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑔) = 0))
3 simp3 1135 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
4 neeq1 3052 . . . . . . . . . . . . 13 (𝑑 = → (𝑑 ≠ 0𝑝 ≠ 0𝑝))
5 fveq2 6649 . . . . . . . . . . . . . 14 (𝑑 = → (deg‘𝑑) = (deg‘))
65breq1d 5043 . . . . . . . . . . . . 13 (𝑑 = → ((deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ↔ (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
7 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑑 = → (coeff‘𝑑) = (coeff‘))
87fveq1d 6651 . . . . . . . . . . . . . . . 16 (𝑑 = → ((coeff‘𝑑)‘𝑒) = ((coeff‘)‘𝑒))
98fveq2d 6653 . . . . . . . . . . . . . . 15 (𝑑 = → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘)‘𝑒)))
109breq1d 5043 . . . . . . . . . . . . . 14 (𝑑 = → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ↔ (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
1110ralbidv 3165 . . . . . . . . . . . . 13 (𝑑 = → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
124, 6, 113anbi123d 1433 . . . . . . . . . . . 12 (𝑑 = → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )) ↔ ( ≠ 0𝑝 ∧ (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))))
13 eldifi 4057 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ∈ (Poly‘ℤ))
1413adantr 484 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
15143adant2 1128 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
16 eldifsni 4686 . . . . . . . . . . . . . . 15 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ≠ 0𝑝)
1716adantr 484 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ≠ 0𝑝)
18 0nn0 11904 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
19 dgrcl 24834 . . . . . . . . . . . . . . . . . . 19 ( ∈ (Poly‘ℤ) → (deg‘) ∈ ℕ0)
2014, 19syl 17 . . . . . . . . . . . . . . . . . 18 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ∈ ℕ0)
21 prssi 4717 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0) → {0, (deg‘)} ⊆ ℕ0)
2218, 20, 21sylancr 590 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘)} ⊆ ℕ0)
23 ssrab2 4010 . . . . . . . . . . . . . . . . . 18 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ ℕ0
2423a1i 11 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ ℕ0)
2522, 24unssd 4116 . . . . . . . . . . . . . . . 16 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℕ0)
26 nn0ssre 11893 . . . . . . . . . . . . . . . . 17 0 ⊆ ℝ
27 ressxr 10678 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℝ*
2826, 27sstri 3927 . . . . . . . . . . . . . . . 16 0 ⊆ ℝ*
2925, 28sstrdi 3930 . . . . . . . . . . . . . . 15 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
30 fvex 6662 . . . . . . . . . . . . . . . . 17 (deg‘) ∈ V
3130prid2 4662 . . . . . . . . . . . . . . . 16 (deg‘) ∈ {0, (deg‘)}
32 elun1 4106 . . . . . . . . . . . . . . . 16 ((deg‘) ∈ {0, (deg‘)} → (deg‘) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
3331, 32ax-mp 5 . . . . . . . . . . . . . . 15 (deg‘) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
34 supxrub 12709 . . . . . . . . . . . . . . 15 ((({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ* ∧ (deg‘) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})) → (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
3529, 33, 34sylancl 589 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
3629adantr 484 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
37 fveq2 6649 . . . . . . . . . . . . . . . . . . . 20 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = (abs‘0))
38 abs0 14641 . . . . . . . . . . . . . . . . . . . 20 (abs‘0) = 0
3937, 38eqtrdi 2852 . . . . . . . . . . . . . . . . . . 19 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = 0)
40 c0ex 10628 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
4140prid1 4661 . . . . . . . . . . . . . . . . . . . 20 0 ∈ {0, (deg‘)}
42 elun1 4106 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {0, (deg‘)} → 0 ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
4341, 42ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
4439, 43eqeltrdi 2901 . . . . . . . . . . . . . . . . . 18 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
4544adantl 485 . . . . . . . . . . . . . . . . 17 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) = 0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
46 eqeq1 2805 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
4746rexbidv 3259 . . . . . . . . . . . . . . . . . . 19 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
48 0z 11984 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
49 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . 24 (coeff‘) = (coeff‘)
5049coef2 24832 . . . . . . . . . . . . . . . . . . . . . . 23 (( ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘):ℕ0⟶ℤ)
5114, 48, 50sylancl 589 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘):ℕ0⟶ℤ)
5251ffvelrnda 6832 . . . . . . . . . . . . . . . . . . . . 21 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘)‘𝑒) ∈ ℤ)
53 nn0abscl 14668 . . . . . . . . . . . . . . . . . . . . 21 (((coeff‘)‘𝑒) ∈ ℤ → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5452, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5554adantr 484 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
56 simplr 768 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
5720ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (deg‘) ∈ ℕ0)
5814ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∈ (Poly‘ℤ))
59 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ((coeff‘)‘𝑒) ≠ 0)
60 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . 23 (deg‘) = (deg‘)
6149, 60dgrub 24835 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
6258, 56, 59, 61syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
63 elfz2nn0 12997 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 ∈ (0...(deg‘)) ↔ (𝑒 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0𝑒 ≤ (deg‘)))
6456, 57, 62, 63syl3anbrc 1340 . . . . . . . . . . . . . . . . . . . 20 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘)))
65 eqid 2801 . . . . . . . . . . . . . . . . . . . 20 (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))
66 2fveq3 6654 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑒 → (abs‘((coeff‘)‘𝑖)) = (abs‘((coeff‘)‘𝑒)))
6766rspceeqv 3589 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 ∈ (0...(deg‘)) ∧ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6864, 65, 67sylancl 589 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6947, 55, 68elrabd 3633 . . . . . . . . . . . . . . . . . 18 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
70 elun2 4107 . . . . . . . . . . . . . . . . . 18 ((abs‘((coeff‘)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
7245, 71pm2.61dane 3077 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
73 supxrub 12709 . . . . . . . . . . . . . . . 16 ((({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ* ∧ (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})) → (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
7436, 72, 73syl2anc 587 . . . . . . . . . . . . . . 15 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
7574ralrimiva 3152 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
7617, 35, 753jca 1125 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ( ≠ 0𝑝 ∧ (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
77763adant2 1128 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ( ≠ 0𝑝 ∧ (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
7812, 15, 77elrabd 3633 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))})
79 simp2 1134 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (𝑔) = 0)
80 fveq1 6648 . . . . . . . . . . . . 13 (𝑐 = → (𝑐𝑔) = (𝑔))
8180eqeq1d 2803 . . . . . . . . . . . 12 (𝑐 = → ((𝑐𝑔) = 0 ↔ (𝑔) = 0))
8281rspcev 3574 . . . . . . . . . . 11 (( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} ∧ (𝑔) = 0) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑔) = 0)
8378, 79, 82syl2anc 587 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑔) = 0)
842, 3, 83elrabd 3633 . . . . . . . . 9 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0})
85 prfi 8781 . . . . . . . . . . . . . 14 {0, (deg‘)} ∈ Fin
86 fzfi 13339 . . . . . . . . . . . . . . . 16 (0...(deg‘)) ∈ Fin
87 abrexfi 8812 . . . . . . . . . . . . . . . 16 ((0...(deg‘)) ∈ Fin → {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin)
8886, 87ax-mp 5 . . . . . . . . . . . . . . 15 {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
89 rabssab 4014 . . . . . . . . . . . . . . 15 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}
90 ssfi 8726 . . . . . . . . . . . . . . 15 (({𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin ∧ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) → {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin)
9188, 89, 90mp2an 691 . . . . . . . . . . . . . 14 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
92 unfi 8773 . . . . . . . . . . . . . 14 (({0, (deg‘)} ∈ Fin ∧ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin)
9385, 91, 92mp2an 691 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin
9433ne0ii 4256 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅
95 xrltso 12526 . . . . . . . . . . . . . 14 < Or ℝ*
96 fisupcl 8921 . . . . . . . . . . . . . 14 (( < Or ℝ* ∧ (({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
9795, 96mpan 689 . . . . . . . . . . . . 13 ((({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
9893, 94, 29, 97mp3an12i 1462 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
9925, 98sseldd 3919 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
100993adant2 1128 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
101 eqidd 2802 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0})
102 breq2 5037 . . . . . . . . . . . . . . 15 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
103 breq2 5037 . . . . . . . . . . . . . . . 16 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
104103ralbidv 3165 . . . . . . . . . . . . . . 15 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < )))
105102, 1043anbi23d 1436 . . . . . . . . . . . . . 14 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))))
106105rabbidv 3430 . . . . . . . . . . . . 13 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))})
107106rexeqdv 3368 . . . . . . . . . . . 12 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0))
108107rabbidv 3430 . . . . . . . . . . 11 (𝑎 = sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0})
109108rspceeqv 3589 . . . . . . . . . 10 ((sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0 ∧ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0}) → ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
110100, 101, 109syl2anc 587 . . . . . . . . 9 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
111 cnex 10611 . . . . . . . . . . 11 ℂ ∈ V
112111rabex 5202 . . . . . . . . . 10 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} ∈ V
113 eleq2 2881 . . . . . . . . . . 11 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} → (𝑔𝑓𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0}))
114 eqeq1 2805 . . . . . . . . . . . 12 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} → (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} ↔ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
115114rexbidv 3259 . . . . . . . . . . 11 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} → (∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} ↔ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
116113, 115anbi12d 633 . . . . . . . . . 10 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} → ((𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}) ↔ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})))
117112, 116spcev 3558 . . . . . . . . 9 ((𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}) → ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
11884, 110, 117syl2anc 587 . . . . . . . 8 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
1191183exp 1116 . . . . . . 7 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ((𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))))
120119rexlimiv 3242 . . . . . 6 (∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})))
121120impcom 411 . . . . 5 ((𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0) → ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
122 eleq2 2881 . . . . . . . . 9 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} → (𝑔𝑓𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
1231rexbidv 3259 . . . . . . . . . . 11 (𝑏 = 𝑔 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0))
124123elrab 3631 . . . . . . . . . 10 (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} ↔ (𝑔 ∈ ℂ ∧ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0))
125 simp1 1133 . . . . . . . . . . . . . . 15 (( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎) → ≠ 0𝑝)
126125anim2i 619 . . . . . . . . . . . . . 14 (( ∈ (Poly‘ℤ) ∧ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)) → ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
1275breq1d 5043 . . . . . . . . . . . . . . . 16 (𝑑 = → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘) ≤ 𝑎))
1289breq1d 5043 . . . . . . . . . . . . . . . . 17 (𝑑 = → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
129128ralbidv 3165 . . . . . . . . . . . . . . . 16 (𝑑 = → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
1304, 127, 1293anbi123d 1433 . . . . . . . . . . . . . . 15 (𝑑 = → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)))
131130elrab 3631 . . . . . . . . . . . . . 14 ( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ↔ ( ∈ (Poly‘ℤ) ∧ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)))
132 eldifsn 4683 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
133126, 131, 1323imtr4i 295 . . . . . . . . . . . . 13 ( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ∈ ((Poly‘ℤ) ∖ {0𝑝}))
134133ssriv 3922 . . . . . . . . . . . 12 {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135 ssrexv 3985 . . . . . . . . . . . . 13 ({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0 → ∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐𝑔) = 0))
13681cbvrexvw 3400 . . . . . . . . . . . . 13 (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐𝑔) = 0 ↔ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)
137135, 136syl6ib 254 . . . . . . . . . . . 12 ({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0 → ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
138134, 137ax-mp 5 . . . . . . . . . . 11 (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0 → ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)
139138anim2i 619 . . . . . . . . . 10 ((𝑔 ∈ ℂ ∧ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑔) = 0) → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
140124, 139sylbi 220 . . . . . . . . 9 (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
141122, 140syl6bi 256 . . . . . . . 8 (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} → (𝑔𝑓 → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)))
142141rexlimivw 3244 . . . . . . 7 (∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} → (𝑔𝑓 → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)))
143142impcom 411 . . . . . 6 ((𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
144143exlimiv 1931 . . . . 5 (∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
145121, 144impbii 212 . . . 4 ((𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0) ↔ ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
146 elaa 24916 . . . 4 (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
147 eluniab 4818 . . . 4 (𝑔 {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}} ↔ ∃𝑓(𝑔𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}))
148145, 146, 1473bitr4i 306 . . 3 (𝑔 ∈ 𝔸 ↔ 𝑔 {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}})
149148eqriv 2798 . 2 𝔸 = {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}}
150 aannenlem.a . . . 4 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
151150rnmpt 5795 . . 3 ran 𝐻 = {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}}
152151unieqi 4816 . 2 ran 𝐻 = {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0}}
153149, 152eqtr4i 2827 1 𝔸 = ran 𝐻
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113   ∖ cdif 3881   ∪ cun 3882   ⊆ wss 3884  ∅c0 4246  {csn 4528  {cpr 4530  ∪ cuni 4803   class class class wbr 5033   ↦ cmpt 5113   Or wor 5441  ran crn 5524  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Fincfn 8496  supcsup 8892  ℂcc 10528  ℝcr 10529  0cc0 10530  ℝ*cxr 10667   < clt 10668   ≤ cle 10669  ℕ0cn0 11889  ℤcz 11973  ...cfz 12889  abscabs 14589  0𝑝c0p 24277  Polycply 24785  coeffccoe 24787  degcdgr 24788  𝔸caa 24914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-rlim 14842  df-sum 15039  df-0p 24278  df-ply 24789  df-coe 24791  df-dgr 24792  df-aa 24915 This theorem is referenced by:  aannenlem3  24930
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