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Theorem aannenlem2 24601
Description: Lemma for aannen 24603. (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 6547 . . . . . . . . . . 11 (𝑏 = 𝑔 → ((𝑐𝑏) = 0 ↔ (𝑐𝑔) = 0))
21rexbidv 3260 . . . . . . . . . 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 1131 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
4 neeq1 3046 . . . . . . . . . . . . 13 (𝑑 = → (𝑑 ≠ 0𝑝 ≠ 0𝑝))
5 fveq2 6538 . . . . . . . . . . . . . 14 (𝑑 = → (deg‘𝑑) = (deg‘))
65breq1d 4972 . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . 17 (𝑑 = → (coeff‘𝑑) = (coeff‘))
87fveq1d 6540 . . . . . . . . . . . . . . . 16 (𝑑 = → ((coeff‘𝑑)‘𝑒) = ((coeff‘)‘𝑒))
98fveq2d 6542 . . . . . . . . . . . . . . 15 (𝑑 = → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘)‘𝑒)))
109breq1d 4972 . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . 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 1428 . . . . . . . . . . . 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 4024 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ∈ (Poly‘ℤ))
1413adantr 481 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
15143adant2 1124 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
16 eldifsni 4629 . . . . . . . . . . . . . . 15 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ≠ 0𝑝)
1716adantr 481 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ≠ 0𝑝)
18 0nn0 11760 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
19 dgrcl 24506 . . . . . . . . . . . . . . . . . . 19 ( ∈ (Poly‘ℤ) → (deg‘) ∈ ℕ0)
2014, 19syl 17 . . . . . . . . . . . . . . . . . 18 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ∈ ℕ0)
21 prssi 4661 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0) → {0, (deg‘)} ⊆ ℕ0)
2218, 20, 21sylancr 587 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘)} ⊆ ℕ0)
23 ssrab2 3977 . . . . . . . . . . . . . . . . . 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 4083 . . . . . . . . . . . . . . . 16 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℕ0)
26 nn0ssre 11749 . . . . . . . . . . . . . . . . 17 0 ⊆ ℝ
27 ressxr 10531 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℝ*
2826, 27sstri 3898 . . . . . . . . . . . . . . . 16 0 ⊆ ℝ*
2925, 28syl6ss 3901 . . . . . . . . . . . . . . 15 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
30 fvex 6551 . . . . . . . . . . . . . . . . 17 (deg‘) ∈ V
3130prid2 4606 . . . . . . . . . . . . . . . 16 (deg‘) ∈ {0, (deg‘)}
32 elun1 4073 . . . . . . . . . . . . . . . 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 12567 . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
3629adantr 481 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
37 fveq2 6538 . . . . . . . . . . . . . . . . . . . 20 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = (abs‘0))
38 abs0 14479 . . . . . . . . . . . . . . . . . . . 20 (abs‘0) = 0
3937, 38syl6eq 2847 . . . . . . . . . . . . . . . . . . 19 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = 0)
40 c0ex 10481 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
4140prid1 4605 . . . . . . . . . . . . . . . . . . . 20 0 ∈ {0, (deg‘)}
42 elun1 4073 . . . . . . . . . . . . . . . . . . . 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, 43syl6eqel 2891 . . . . . . . . . . . . . . . . . 18 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
4544adantl 482 . . . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
4746rexbidv 3260 . . . . . . . . . . . . . . . . . . 19 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
48 0z 11840 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
49 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . . 24 (coeff‘) = (coeff‘)
5049coef2 24504 . . . . . . . . . . . . . . . . . . . . . . 23 (( ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘):ℕ0⟶ℤ)
5114, 48, 50sylancl 586 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘):ℕ0⟶ℤ)
5251ffvelrnda 6716 . . . . . . . . . . . . . . . . . . . . 21 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘)‘𝑒) ∈ ℤ)
53 nn0abscl 14506 . . . . . . . . . . . . . . . . . . . . 21 (((coeff‘)‘𝑒) ∈ ℤ → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5452, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5554adantr 481 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
56 simplr 765 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
5720ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (deg‘) ∈ ℕ0)
5814ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∈ (Poly‘ℤ))
59 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ((coeff‘)‘𝑒) ≠ 0)
60 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . 23 (deg‘) = (deg‘)
6149, 60dgrub 24507 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
6258, 56, 59, 61syl3anc 1364 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
63 elfz2nn0 12848 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 ∈ (0...(deg‘)) ↔ (𝑒 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0𝑒 ≤ (deg‘)))
6456, 57, 62, 63syl3anbrc 1336 . . . . . . . . . . . . . . . . . . . 20 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘)))
65 eqid 2795 . . . . . . . . . . . . . . . . . . . 20 (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))
66 2fveq3 6543 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑒 → (abs‘((coeff‘)‘𝑖)) = (abs‘((coeff‘)‘𝑒)))
6766rspceeqv 3577 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 ∈ (0...(deg‘)) ∧ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6864, 65, 67sylancl 586 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6947, 55, 68elrabd 3620 . . . . . . . . . . . . . . . . . 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 4074 . . . . . . . . . . . . . . . . . 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 3072 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
73 supxrub 12567 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . 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 1121 . . . . . . . . . . . . 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 1124 . . . . . . . . . . . 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 3620 . . . . . . . . . . 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 1130 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (𝑔) = 0)
80 fveq1 6537 . . . . . . . . . . . . 13 (𝑐 = → (𝑐𝑔) = (𝑔))
8180eqeq1d 2797 . . . . . . . . . . . 12 (𝑐 = → ((𝑐𝑔) = 0 ↔ (𝑔) = 0))
8281rspcev 3559 . . . . . . . . . . 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 584 . . . . . . . . . 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 3620 . . . . . . . . 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 8639 . . . . . . . . . . . . . 14 {0, (deg‘)} ∈ Fin
86 fzfi 13190 . . . . . . . . . . . . . . . 16 (0...(deg‘)) ∈ Fin
87 abrexfi 8670 . . . . . . . . . . . . . . . 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 3981 . . . . . . . . . . . . . . 15 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}
90 ssfi 8584 . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . 14 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
92 unfi 8631 . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin
9433ne0ii 4223 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅
95 xrltso 12384 . . . . . . . . . . . . . 14 < Or ℝ*
96 fisupcl 8779 . . . . . . . . . . . . . 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 686 . . . . . . . . . . . . 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 1457 . . . . . . . . . . . 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 3890 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
100993adant2 1124 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
101 eqidd 2796 . . . . . . . . . 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 4966 . . . . . . . . . . . . . . 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 4966 . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . 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 1431 . . . . . . . . . . . . . 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 3425 . . . . . . . . . . . . 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 3376 . . . . . . . . . . . 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 3425 . . . . . . . . . . 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 3577 . . . . . . . . . 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 584 . . . . . . . . 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 10464 . . . . . . . . . . 11 ℂ ∈ V
112111rabex 5126 . . . . . . . . . 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 2871 . . . . . . . . . . 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 2799 . . . . . . . . . . . 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 3260 . . . . . . . . . . 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 630 . . . . . . . . . 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 3549 . . . . . . . . 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 584 . . . . . . . 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 1112 . . . . . . 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 3243 . . . . . 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 408 . . . . 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 2871 . . . . . . . . 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 3260 . . . . . . . . . . 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 3618 . . . . . . . . . 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 1129 . . . . . . . . . . . . . . 15 (( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎) → ≠ 0𝑝)
126125anim2i 616 . . . . . . . . . . . . . 14 (( ∈ (Poly‘ℤ) ∧ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)) → ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
1275breq1d 4972 . . . . . . . . . . . . . . . 16 (𝑑 = → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘) ≤ 𝑎))
1289breq1d 4972 . . . . . . . . . . . . . . . . 17 (𝑑 = → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
129128ralbidv 3164 . . . . . . . . . . . . . . . 16 (𝑑 = → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
1304, 127, 1293anbi123d 1428 . . . . . . . . . . . . . . 15 (𝑑 = → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)))
131130elrab 3618 . . . . . . . . . . . . . 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 4626 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
133126, 131, 1323imtr4i 293 . . . . . . . . . . . . 13 ( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ∈ ((Poly‘ℤ) ∖ {0𝑝}))
134133ssriv 3893 . . . . . . . . . . . 12 {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135 ssrexv 3955 . . . . . . . . . . . . 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))
13681cbvrexv 3404 . . . . . . . . . . . . 13 (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐𝑔) = 0 ↔ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)
137135, 136syl6ib 252 . . . . . . . . . . . 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 616 . . . . . . . . . 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 218 . . . . . . . . 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 254 . . . . . . . 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 3245 . . . . . . 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 408 . . . . . 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 1908 . . . . 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 210 . . . 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 24588 . . . 4 (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
147 eluniab 4756 . . . 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 304 . . 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 2792 . 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 5709 . . 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 4754 . 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 2822 1 𝔸 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  {cab 2775  wne 2984  wral 3105  wrex 3106  {crab 3109  cdif 3856  cun 3857  wss 3859  c0 4211  {csn 4472  {cpr 4474   cuni 4745   class class class wbr 4962  cmpt 5041   Or wor 5361  ran crn 5444  wf 6221  cfv 6225  (class class class)co 7016  Fincfn 8357  supcsup 8750  cc 10381  cr 10382  0cc0 10383  *cxr 10520   < clt 10521  cle 10522  0cn0 11745  cz 11829  ...cfz 12742  abscabs 14427  0𝑝c0p 23953  Polycply 24457  coeffccoe 24459  degcdgr 24460  𝔸caa 24586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-0p 23954  df-ply 24461  df-coe 24463  df-dgr 24464  df-aa 24587
This theorem is referenced by:  aannenlem3  24602
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