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Theorem aannenlem2 26349
Description: Lemma for aannen 26351. (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 6909 . . . . . . . . . . 11 (𝑏 = 𝑔 → ((𝑐𝑏) = 0 ↔ (𝑐𝑔) = 0))
21rexbidv 3168 . . . . . . . . . 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 2992 . . . . . . . . . . . . 13 (𝑑 = → (𝑑 ≠ 0𝑝 ≠ 0𝑝))
5 fveq2 6900 . . . . . . . . . . . . . 14 (𝑑 = → (deg‘𝑑) = (deg‘))
65breq1d 5162 . . . . . . . . . . . . 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 6900 . . . . . . . . . . . . . . . . 17 (𝑑 = → (coeff‘𝑑) = (coeff‘))
87fveq1d 6902 . . . . . . . . . . . . . . . 16 (𝑑 = → ((coeff‘𝑑)‘𝑒) = ((coeff‘)‘𝑒))
98fveq2d 6904 . . . . . . . . . . . . . . 15 (𝑑 = → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘)‘𝑒)))
109breq1d 5162 . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . 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 1432 . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ∈ (Poly‘ℤ))
1413adantr 479 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
15143adant2 1128 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
16 eldifsni 4798 . . . . . . . . . . . . . . 15 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ≠ 0𝑝)
1716adantr 479 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ≠ 0𝑝)
18 0nn0 12534 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
19 dgrcl 26252 . . . . . . . . . . . . . . . . . . 19 ( ∈ (Poly‘ℤ) → (deg‘) ∈ ℕ0)
2014, 19syl 17 . . . . . . . . . . . . . . . . . 18 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ∈ ℕ0)
21 prssi 4829 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0) → {0, (deg‘)} ⊆ ℕ0)
2218, 20, 21sylancr 585 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘)} ⊆ ℕ0)
23 ssrab2 4075 . . . . . . . . . . . . . . . . . 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 4186 . . . . . . . . . . . . . . . 16 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℕ0)
26 nn0ssre 12523 . . . . . . . . . . . . . . . . 17 0 ⊆ ℝ
27 ressxr 11304 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℝ*
2826, 27sstri 3988 . . . . . . . . . . . . . . . 16 0 ⊆ ℝ*
2925, 28sstrdi 3991 . . . . . . . . . . . . . . 15 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
30 fvex 6913 . . . . . . . . . . . . . . . . 17 (deg‘) ∈ V
3130prid2 4771 . . . . . . . . . . . . . . . 16 (deg‘) ∈ {0, (deg‘)}
32 elun1 4176 . . . . . . . . . . . . . . . 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 13352 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
3629adantr 479 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
37 fveq2 6900 . . . . . . . . . . . . . . . . . . . 20 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = (abs‘0))
38 abs0 15285 . . . . . . . . . . . . . . . . . . . 20 (abs‘0) = 0
3937, 38eqtrdi 2781 . . . . . . . . . . . . . . . . . . 19 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = 0)
40 c0ex 11254 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
4140prid1 4770 . . . . . . . . . . . . . . . . . . . 20 0 ∈ {0, (deg‘)}
42 elun1 4176 . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . 18 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
4544adantl 480 . . . . . . . . . . . . . . . . 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 2729 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
4746rexbidv 3168 . . . . . . . . . . . . . . . . . . 19 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
48 0z 12616 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
49 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . 24 (coeff‘) = (coeff‘)
5049coef2 26250 . . . . . . . . . . . . . . . . . . . . . . 23 (( ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘):ℕ0⟶ℤ)
5114, 48, 50sylancl 584 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘):ℕ0⟶ℤ)
5251ffvelcdmda 7097 . . . . . . . . . . . . . . . . . . . . 21 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘)‘𝑒) ∈ ℤ)
53 nn0abscl 15312 . . . . . . . . . . . . . . . . . . . . 21 (((coeff‘)‘𝑒) ∈ ℤ → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5452, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
5554adantr 479 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
56 simplr 767 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
5720ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (deg‘) ∈ ℕ0)
5814ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∈ (Poly‘ℤ))
59 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ((coeff‘)‘𝑒) ≠ 0)
60 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . 23 (deg‘) = (deg‘)
6149, 60dgrub 26253 . . . . . . . . . . . . . . . . . . . . . 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 13641 . . . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . . . 20 (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))
66 2fveq3 6905 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑒 → (abs‘((coeff‘)‘𝑖)) = (abs‘((coeff‘)‘𝑒)))
6766rspceeqv 3629 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 ∈ (0...(deg‘)) ∧ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6864, 65, 67sylancl 584 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
6947, 55, 68elrabd 3682 . . . . . . . . . . . . . . . . . 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 4177 . . . . . . . . . . . . . . . . . 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 3018 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
73 supxrub 13352 . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . 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 3682 . . . . . . . . . . 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 6899 . . . . . . . . . . . . 13 (𝑐 = → (𝑐𝑔) = (𝑔))
8180eqeq1d 2727 . . . . . . . . . . . 12 (𝑐 = → ((𝑐𝑔) = 0 ↔ (𝑔) = 0))
8281rspcev 3607 . . . . . . . . . . 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 582 . . . . . . . . . 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 3682 . . . . . . . . 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 9360 . . . . . . . . . . . . . 14 {0, (deg‘)} ∈ Fin
86 fzfi 13987 . . . . . . . . . . . . . . . 16 (0...(deg‘)) ∈ Fin
87 abrexfi 9392 . . . . . . . . . . . . . . . 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 4081 . . . . . . . . . . . . . . 15 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}
90 ssfi 9210 . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . 14 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
92 unfi 9209 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin
9433ne0ii 4339 . . . . . . . . . . . . 13 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅
95 xrltso 13169 . . . . . . . . . . . . . 14 < Or ℝ*
96 fisupcl 9508 . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . 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 1461 . . . . . . . . . . . 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 3979 . . . . . . . . . . 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 2726 . . . . . . . . . 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 5156 . . . . . . . . . . . . . . 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 5156 . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . 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 1435 . . . . . . . . . . . . . 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 3426 . . . . . . . . . . . . 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 3315 . . . . . . . . . . . 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 3426 . . . . . . . . . . 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 3629 . . . . . . . . . 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 582 . . . . . . . . 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 11235 . . . . . . . . . . 11 ℂ ∈ V
112111rabex 5338 . . . . . . . . . 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 2814 . . . . . . . . . . 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 2729 . . . . . . . . . . . 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 3168 . . . . . . . . . . 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 3591 . . . . . . . . 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 582 . . . . . . . 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 3137 . . . . . 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 406 . . . . 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 2814 . . . . . . . . 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 3168 . . . . . . . . . . 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 3680 . . . . . . . . . 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 615 . . . . . . . . . . . . . 14 (( ∈ (Poly‘ℤ) ∧ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)) → ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
1275breq1d 5162 . . . . . . . . . . . . . . . 16 (𝑑 = → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘) ≤ 𝑎))
1289breq1d 5162 . . . . . . . . . . . . . . . . 17 (𝑑 = → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
129128ralbidv 3167 . . . . . . . . . . . . . . . 16 (𝑑 = → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
1304, 127, 1293anbi123d 1432 . . . . . . . . . . . . . . 15 (𝑑 = → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)))
131130elrab 3680 . . . . . . . . . . . . . 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 4794 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
133126, 131, 1323imtr4i 291 . . . . . . . . . . . . 13 ( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ∈ ((Poly‘ℤ) ∖ {0𝑝}))
134133ssriv 3982 . . . . . . . . . . . 12 {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135 ssrexv 4048 . . . . . . . . . . . . 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 3225 . . . . . . . . . . . . 13 (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐𝑔) = 0 ↔ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)
137135, 136imbitrdi 250 . . . . . . . . . . . 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 615 . . . . . . . . . 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 216 . . . . . . . . 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, 140biimtrdi 252 . . . . . . . 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 3140 . . . . . . 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 406 . . . . . 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 1925 . . . . 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 208 . . . 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 26336 . . . 4 (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
147 eluniab 4926 . . . 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 302 . . 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 2722 . 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 5960 . . 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 4924 . 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 2756 1 𝔸 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wne 2929  wral 3050  wrex 3059  {crab 3418  cdif 3943  cun 3944  wss 3946  c0 4324  {csn 4632  {cpr 4634   cuni 4912   class class class wbr 5152  cmpt 5235   Or wor 5592  ran crn 5682  wf 6549  cfv 6553  (class class class)co 7423  Fincfn 8973  supcsup 9479  cc 11152  cr 11153  0cc0 11154  *cxr 11293   < clt 11294  cle 11295  0cn0 12519  cz 12605  ...cfz 13533  abscabs 15234  0𝑝c0p 25681  Polycply 26203  coeffccoe 26205  degcdgr 26206  𝔸caa 26334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-inf2 9680  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231  ax-pre-sup 11232
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7689  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-er 8733  df-map 8856  df-pm 8857  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9481  df-inf 9482  df-oi 9549  df-card 9978  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-div 11918  df-nn 12260  df-2 12322  df-3 12323  df-n0 12520  df-z 12606  df-uz 12870  df-rp 13024  df-fz 13534  df-fzo 13677  df-fl 13807  df-seq 14017  df-exp 14077  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-clim 15485  df-rlim 15486  df-sum 15686  df-0p 25682  df-ply 26207  df-coe 26209  df-dgr 26210  df-aa 26335
This theorem is referenced by:  aannenlem3  26350
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