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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  cc 10528  cr 10529  0cc0 10530  *cxr 10667   < clt 10668  cle 10669  0cn0 11889  cz 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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