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Theorem aannenlem2 24303
Description: Lemma for aannen 24305. (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 simp3 1133 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
2 eldifi 3875 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ∈ (Poly‘ℤ))
32adantr 472 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
433adant2 1126 . . . . . . . . . . . 12 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∈ (Poly‘ℤ))
5 eldifsni 4466 . . . . . . . . . . . . . . 15 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ≠ 0𝑝)
65adantr 472 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ≠ 0𝑝)
7 0nn0 11519 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
8 dgrcl 24208 . . . . . . . . . . . . . . . . . . 19 ( ∈ (Poly‘ℤ) → (deg‘) ∈ ℕ0)
93, 8syl 17 . . . . . . . . . . . . . . . . . 18 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ∈ ℕ0)
10 prssi 4498 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0) → {0, (deg‘)} ⊆ ℕ0)
117, 9, 10sylancr 698 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘)} ⊆ ℕ0)
12 ssrab2 3828 . . . . . . . . . . . . . . . . . 18 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ ℕ0
1312a1i 11 . . . . . . . . . . . . . . . . 17 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ ℕ0)
1411, 13unssd 3932 . . . . . . . . . . . . . . . 16 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℕ0)
15 nn0ssre 11508 . . . . . . . . . . . . . . . . 17 0 ⊆ ℝ
16 ressxr 10295 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℝ*
1715, 16sstri 3753 . . . . . . . . . . . . . . . 16 0 ⊆ ℝ*
1814, 17syl6ss 3756 . . . . . . . . . . . . . . 15 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
19 fvex 6363 . . . . . . . . . . . . . . . . 17 (deg‘) ∈ V
2019prid2 4442 . . . . . . . . . . . . . . . 16 (deg‘) ∈ {0, (deg‘)}
21 elun1 3923 . . . . . . . . . . . . . . . 16 ((deg‘) ∈ {0, (deg‘)} → (deg‘) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 (deg‘) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
23 supxrub 12367 . . . . . . . . . . . . . . 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*, < ))
2418, 22, 23sylancl 697 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
2518adantr 472 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ⊆ ℝ*)
26 fveq2 6353 . . . . . . . . . . . . . . . . . . . 20 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = (abs‘0))
27 abs0 14244 . . . . . . . . . . . . . . . . . . . 20 (abs‘0) = 0
2826, 27syl6eq 2810 . . . . . . . . . . . . . . . . . . 19 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) = 0)
29 c0ex 10246 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
3029prid1 4441 . . . . . . . . . . . . . . . . . . . 20 0 ∈ {0, (deg‘)}
31 elun1 3923 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {0, (deg‘)} → 0 ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
3328, 32syl6eqel 2847 . . . . . . . . . . . . . . . . . 18 (((coeff‘)‘𝑒) = 0 → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
3433adantl 473 . . . . . . . . . . . . . . . . 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))}))
35 0z 11600 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
36 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . 24 (coeff‘) = (coeff‘)
3736coef2 24206 . . . . . . . . . . . . . . . . . . . . . . 23 (( ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘):ℕ0⟶ℤ)
383, 35, 37sylancl 697 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘):ℕ0⟶ℤ)
3938ffvelrnda 6523 . . . . . . . . . . . . . . . . . . . . 21 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘)‘𝑒) ∈ ℤ)
40 nn0abscl 14271 . . . . . . . . . . . . . . . . . . . . 21 (((coeff‘)‘𝑒) ∈ ℤ → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
4139, 40syl 17 . . . . . . . . . . . . . . . . . . . 20 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
4241adantr 472 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ ℕ0)
43 simplr 809 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
449ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (deg‘) ∈ ℕ0)
453ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∈ (Poly‘ℤ))
46 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ((coeff‘)‘𝑒) ≠ 0)
47 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . 23 (deg‘) = (deg‘)
4836, 47dgrub 24209 . . . . . . . . . . . . . . . . . . . . . 22 (( ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
4945, 43, 46, 48syl3anc 1477 . . . . . . . . . . . . . . . . . . . . 21 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘))
50 elfz2nn0 12644 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 ∈ (0...(deg‘)) ↔ (𝑒 ∈ ℕ0 ∧ (deg‘) ∈ ℕ0𝑒 ≤ (deg‘)))
5143, 44, 49, 50syl3anbrc 1429 . . . . . . . . . . . . . . . . . . . 20 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘)))
52 eqid 2760 . . . . . . . . . . . . . . . . . . . 20 (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))
53 fveq2 6353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑒 → ((coeff‘)‘𝑖) = ((coeff‘)‘𝑒))
5453fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑒 → (abs‘((coeff‘)‘𝑖)) = (abs‘((coeff‘)‘𝑒)))
5554eqeq2d 2770 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑒 → ((abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)) ↔ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))))
5655rspcev 3449 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 ∈ (0...(deg‘)) ∧ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
5751, 52, 56sylancl 697 . . . . . . . . . . . . . . . . . . 19 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖)))
58 eqeq1 2764 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ (abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
5958rexbidv 3190 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (abs‘((coeff‘)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
6059elrab 3504 . . . . . . . . . . . . . . . . . . 19 ((abs‘((coeff‘)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ↔ ((abs‘((coeff‘)‘𝑒)) ∈ ℕ0 ∧ ∃𝑖 ∈ (0...(deg‘))(abs‘((coeff‘)‘𝑒)) = (abs‘((coeff‘)‘𝑖))))
6142, 57, 60sylanbrc 701 . . . . . . . . . . . . . . . . . 18 (((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘)‘𝑒) ≠ 0) → (abs‘((coeff‘)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))})
62 elun2 3924 . . . . . . . . . . . . . . . . . 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))}))
6361, 62syl 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))}))
6434, 63pm2.61dane 3019 . . . . . . . . . . . . . . . 16 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ∈ ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}))
65 supxrub 12367 . . . . . . . . . . . . . . . 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*, < ))
6625, 64, 65syl2anc 696 . . . . . . . . . . . . . . 15 ((( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
6766ralrimiva 3104 . . . . . . . . . . . . . 14 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ))
686, 24, 673jca 1123 . . . . . . . . . . . . 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*, < )))
69683adant2 1126 . . . . . . . . . . . 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*, < )))
70 neeq1 2994 . . . . . . . . . . . . . 14 (𝑑 = → (𝑑 ≠ 0𝑝 ≠ 0𝑝))
71 fveq2 6353 . . . . . . . . . . . . . . 15 (𝑑 = → (deg‘𝑑) = (deg‘))
7271breq1d 4814 . . . . . . . . . . . . . 14 (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*, < )))
73 fveq2 6353 . . . . . . . . . . . . . . . . . 18 (𝑑 = → (coeff‘𝑑) = (coeff‘))
7473fveq1d 6355 . . . . . . . . . . . . . . . . 17 (𝑑 = → ((coeff‘𝑑)‘𝑒) = ((coeff‘)‘𝑒))
7574fveq2d 6357 . . . . . . . . . . . . . . . 16 (𝑑 = → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘)‘𝑒)))
7675breq1d 4814 . . . . . . . . . . . . . . 15 (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*, < )))
7776ralbidv 3124 . . . . . . . . . . . . . 14 (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*, < )))
7870, 72, 773anbi123d 1548 . . . . . . . . . . . . 13 (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*, < ))))
7978elrab 3504 . . . . . . . . . . . 12 ( ∈ {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*, < ))} ↔ ( ∈ (Poly‘ℤ) ∧ ( ≠ 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*, < ))))
804, 69, 79sylanbrc 701 . . . . . . . . . . 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*, < ))})
81 simp2 1132 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (𝑔) = 0)
82 fveq1 6352 . . . . . . . . . . . . 13 (𝑐 = → (𝑐𝑔) = (𝑔))
8382eqeq1d 2762 . . . . . . . . . . . 12 (𝑐 = → ((𝑐𝑔) = 0 ↔ (𝑔) = 0))
8483rspcev 3449 . . . . . . . . . . 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)
8580, 81, 84syl2anc 696 . . . . . . . . . 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)
86 fveq2 6353 . . . . . . . . . . . . 13 (𝑏 = 𝑔 → (𝑐𝑏) = (𝑐𝑔))
8786eqeq1d 2762 . . . . . . . . . . . 12 (𝑏 = 𝑔 → ((𝑐𝑏) = 0 ↔ (𝑐𝑔) = 0))
8887rexbidv 3190 . . . . . . . . . . 11 (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))
8988elrab 3504 . . . . . . . . . 10 (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} ↔ (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))
901, 85, 89sylanbrc 701 . . . . . . . . 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})
91 prfi 8402 . . . . . . . . . . . . . . 15 {0, (deg‘)} ∈ Fin
92 fzfi 12985 . . . . . . . . . . . . . . . . 17 (0...(deg‘)) ∈ Fin
93 abrexfi 8433 . . . . . . . . . . . . . . . . 17 ((0...(deg‘)) ∈ Fin → {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin)
9492, 93ax-mp 5 . . . . . . . . . . . . . . . 16 {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
95 rabssab 3832 . . . . . . . . . . . . . . . 16 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}
96 ssfi 8347 . . . . . . . . . . . . . . . 16 (({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)
9794, 95, 96mp2an 710 . . . . . . . . . . . . . . 15 {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))} ∈ Fin
98 unfi 8394 . . . . . . . . . . . . . . 15 (({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)
9991, 97, 98mp2an 710 . . . . . . . . . . . . . 14 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin
10099a1i 11 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ∈ Fin)
10122ne0ii 4066 . . . . . . . . . . . . . 14 ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅
102101a1i 11 . . . . . . . . . . . . 13 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}) ≠ ∅)
103 xrltso 12187 . . . . . . . . . . . . . 14 < Or ℝ*
104 fisupcl 8542 . . . . . . . . . . . . . 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))}))
105103, 104mpan 708 . . . . . . . . . . . . 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))}))
106100, 102, 18, 105syl3anc 1477 . . . . . . . . . . . 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))}))
10714, 106sseldd 3745 . . . . . . . . . . 11 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
1081073adant2 1126 . . . . . . . . . 10 (( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑔) = 0 ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘))𝑔 = (abs‘((coeff‘)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
109 eqidd 2761 . . . . . . . . . 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})
110 breq2 4808 . . . . . . . . . . . . . . . 16 (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*, < )))
111 breq2 4808 . . . . . . . . . . . . . . . . 17 (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*, < )))
112111ralbidv 3124 . . . . . . . . . . . . . . . 16 (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*, < )))
113110, 1123anbi23d 1551 . . . . . . . . . . . . . . 15 (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*, < ))))
114113rabbidv 3329 . . . . . . . . . . . . . 14 (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*, < ))})
115114rexeqdv 3284 . . . . . . . . . . . . 13 (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))
116115rabbidv 3329 . . . . . . . . . . . 12 (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})
117116eqeq2d 2770 . . . . . . . . . . 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) ≤ 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} ↔ {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}))
118117rspcev 3449 . . . . . . . . . 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})
119108, 109, 118syl2anc 696 . . . . . . . . 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})
120 cnex 10229 . . . . . . . . . . 11 ℂ ∈ V
121120rabex 4964 . . . . . . . . . 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
122 eleq2 2828 . . . . . . . . . . 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}))
123 eqeq1 2764 . . . . . . . . . . . 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}))
124123rexbidv 3190 . . . . . . . . . . 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}))
125122, 124anbi12d 749 . . . . . . . . . 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})))
126121, 125spcev 3440 . . . . . . . . 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}))
12790, 119, 126syl2anc 696 . . . . . . . 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}))
1281273exp 1113 . . . . . . 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}))))
129128rexlimiv 3165 . . . . . 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})))
130129impcom 445 . . . . 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}))
131 eleq2 2828 . . . . . . . . 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}))
13287rexbidv 3190 . . . . . . . . . . 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))
133132elrab 3504 . . . . . . . . . 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))
134 simp1 1131 . . . . . . . . . . . . . . 15 (( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎) → ≠ 0𝑝)
135134anim2i 594 . . . . . . . . . . . . . 14 (( ∈ (Poly‘ℤ) ∧ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)) → ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
13671breq1d 4814 . . . . . . . . . . . . . . . 16 (𝑑 = → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘) ≤ 𝑎))
13775breq1d 4814 . . . . . . . . . . . . . . . . 17 (𝑑 = → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
138137ralbidv 3124 . . . . . . . . . . . . . . . 16 (𝑑 = → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎))
13970, 136, 1383anbi123d 1548 . . . . . . . . . . . . . . 15 (𝑑 = → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ ( ≠ 0𝑝 ∧ (deg‘) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘)‘𝑒)) ≤ 𝑎)))
140139elrab 3504 . . . . . . . . . . . . . 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)))
141 eldifsn 4462 . . . . . . . . . . . . . 14 ( ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ ( ∈ (Poly‘ℤ) ∧ ≠ 0𝑝))
142135, 140, 1413imtr4i 281 . . . . . . . . . . . . 13 ( ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ∈ ((Poly‘ℤ) ∖ {0𝑝}))
143142ssriv 3748 . . . . . . . . . . . 12 {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
144 ssrexv 3808 . . . . . . . . . . . . 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))
14583cbvrexv 3311 . . . . . . . . . . . . 13 (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐𝑔) = 0 ↔ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0)
146144, 145syl6ib 241 . . . . . . . . . . . 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))
147143, 146ax-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)
148147anim2i 594 . . . . . . . . . 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))
149133, 148sylbi 207 . . . . . . . . 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))
150131, 149syl6bi 243 . . . . . . . 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)))
151150rexlimivw 3167 . . . . . . 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)))
152151impcom 445 . . . . . 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))
153152exlimiv 2007 . . . . 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))
154130, 153impbii 199 . . . 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}))
155 elaa 24290 . . . 4 (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔) = 0))
156 eluniab 4599 . . . 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}))
157154, 155, 1563bitr4i 292 . . 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}})
158157eqriv 2757 . 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}}
159 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})
160159rnmpt 5526 . . 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}}
161160unieqi 4597 . 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}}
162158, 161eqtr4i 2785 1 𝔸 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  cdif 3712  cun 3713  wss 3715  c0 4058  {csn 4321  {cpr 4323   cuni 4588   class class class wbr 4804  cmpt 4881   Or wor 5186  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6814  Fincfn 8123  supcsup 8513  cc 10146  cr 10147  0cc0 10148  *cxr 10285   < clt 10286  cle 10287  0cn0 11504  cz 11589  ...cfz 12539  abscabs 14193  0𝑝c0p 23655  Polycply 24159  coeffccoe 24161  degcdgr 24162  𝔸caa 24288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-0p 23656  df-ply 24163  df-coe 24165  df-dgr 24166  df-aa 24289
This theorem is referenced by:  aannenlem3  24304
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