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Theorem mpaaeu 36563
Description: An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaeu (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
Distinct variable group:   𝐴,𝑝

Proof of Theorem mpaaeu
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsscn 11634 . . . . . 6 ℚ ⊆ ℂ
2 eldifi 3693 . . . . . . . . . 10 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
32ad2antlr 758 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4 zssq 11630 . . . . . . . . . 10 ℤ ⊆ ℚ
5 0z 11224 . . . . . . . . . 10 0 ∈ ℤ
64, 5sselii 3564 . . . . . . . . 9 0 ∈ ℚ
7 eqid 2609 . . . . . . . . . 10 (coeff‘𝑎) = (coeff‘𝑎)
87coef2 23736 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
93, 6, 8sylancl 692 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10 dgrcl 23738 . . . . . . . . 9 (𝑎 ∈ (Poly‘ℚ) → (deg‘𝑎) ∈ ℕ0)
113, 10syl 17 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘𝑎) ∈ ℕ0)
129, 11ffvelrnd 6253 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13 eldifsni 4260 . . . . . . . . 9 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
1413ad2antlr 758 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15 eqid 2609 . . . . . . . . . . 11 (deg‘𝑎) = (deg‘𝑎)
1615, 7dgreq0 23770 . . . . . . . . . 10 (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
1716necon3bid 2825 . . . . . . . . 9 (𝑎 ∈ (Poly‘ℚ) → (𝑎 ≠ 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0))
183, 17syl 17 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (𝑎 ≠ 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0))
1914, 18mpbid 220 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20 qreccl 11643 . . . . . . 7 ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
2112, 19, 20syl2anc 690 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22 plyconst 23711 . . . . . 6 ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
231, 21, 22sylancr 693 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24 simpl 471 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25 simpr 475 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26 qaddcl 11639 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
2726adantl 480 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28 qmulcl 11641 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
2928adantl 480 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
3024, 25, 27, 29plymul 23723 . . . . 5 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) ∈ (Poly‘ℚ))
3123, 3, 30syl2anc 690 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) ∈ (Poly‘ℚ))
327coef3 23737 . . . . . . . . 9 (𝑎 ∈ (Poly‘ℚ) → (coeff‘𝑎):ℕ0⟶ℂ)
333, 32syl 17 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℂ)
3433, 11ffvelrnd 6253 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
3534, 19reccld 10646 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
3634, 19recne0d 10647 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37 dgrmulc 23776 . . . . . 6 (((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0 ∧ 𝑎 ∈ (Poly‘ℚ)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (deg‘𝑎))
3835, 36, 3, 37syl3anc 1317 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (deg‘𝑎))
39 simprl 789 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘𝑎) = (degAA𝐴))
4038, 39eqtrd 2643 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (degAA𝐴))
41 aacn 23821 . . . . . . 7 (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
4241ad2antrr 757 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝐴 ∈ ℂ)
43 ovex 6555 . . . . . . . 8 (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44 fnconstg 5991 . . . . . . . 8 ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℂ)
4543, 44mp1i 13 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℂ)
46 plyf 23703 . . . . . . . 8 (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47 ffn 5944 . . . . . . . 8 (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
483, 46, 473syl 18 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 Fn ℂ)
49 cnex 9874 . . . . . . . 8 ℂ ∈ V
5049a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
51 inidm 3783 . . . . . . 7 (ℂ ∩ ℂ) = ℂ
5243fvconst2 6352 . . . . . . . 8 (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
5352adantl 480 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
54 simplrr 796 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
5545, 48, 50, 50, 51, 53, 54ofval 6782 . . . . . 6 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0))
5642, 55mpdan 698 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0))
5735mul01d 10087 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0) = 0)
5856, 57eqtrd 2643 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0)
59 coemulc 23760 . . . . . . 7 (((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ ∧ 𝑎 ∈ (Poly‘ℚ)) → (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · (coeff‘𝑎)))
6035, 3, 59syl2anc 690 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · (coeff‘𝑎)))
6160fveq1d 6090 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)) = (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · (coeff‘𝑎))‘(degAA𝐴)))
62 dgraacl 36559 . . . . . . . 8 (𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
6362ad2antrr 757 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ)
6463nnnn0d 11201 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ0)
65 fnconstg 5991 . . . . . . . 8 ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V → (ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℕ0)
6643, 65mp1i 13 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℕ0)
67 ffn 5944 . . . . . . . 8 ((coeff‘𝑎):ℕ0⟶ℂ → (coeff‘𝑎) Fn ℕ0)
6833, 67syl 17 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
69 nn0ex 11148 . . . . . . . 8 0 ∈ V
7069a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℕ0 ∈ V)
71 inidm 3783 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
7243fvconst2 6352 . . . . . . . 8 ((degAA𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
7372adantl 480 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
74 simplrl 795 . . . . . . . . 9 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA𝐴))
7574eqcomd 2615 . . . . . . . 8 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (degAA𝐴) = (deg‘𝑎))
7675fveq2d 6092 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → ((coeff‘𝑎)‘(degAA𝐴)) = ((coeff‘𝑎)‘(deg‘𝑎)))
7766, 68, 70, 70, 71, 73, 76ofval 6782 . . . . . 6 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · (coeff‘𝑎))‘(degAA𝐴)) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))))
7864, 77mpdan 698 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · (coeff‘𝑎))‘(degAA𝐴)) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))))
7934, 19recid2d 10649 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))) = 1)
8061, 78, 793eqtrd 2647 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)) = 1)
81 fveq2 6088 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (deg‘𝑝) = (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)))
8281eqeq1d 2611 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (degAA𝐴)))
83 fveq1 6087 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (𝑝𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴))
8483eqeq1d 2611 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((𝑝𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0))
85 fveq2 6088 . . . . . . . 8 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)))
8685fveq1d 6090 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)))
8786eqeq1d 2611 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)) = 1))
8882, 84, 873anbi123d 1390 . . . . 5 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (degAA𝐴) ∧ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0 ∧ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)) = 1)))
8988rspcev 3281 . . . 4 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) ∈ (Poly‘ℚ) ∧ ((deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (degAA𝐴) ∧ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0 ∧ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)) = 1)) → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
9031, 40, 58, 80, 89syl13anc 1319 . . 3 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
91 dgraalem 36558 . . . 4 (𝐴 ∈ 𝔸 → ((degAA𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)))
9291simprd 477 . . 3 (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0))
9390, 92r19.29a 3059 . 2 (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
94 simp2 1054 . . . . . . . . . . 11 (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝𝐴) = 0)
95 simp2 1054 . . . . . . . . . . 11 (((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1) → (𝑎𝐴) = 0)
9694, 95anim12i 587 . . . . . . . . . 10 ((((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)) → ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0))
97 plyf 23703 . . . . . . . . . . . . . . . 16 (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
98 ffn 5944 . . . . . . . . . . . . . . . 16 (𝑝:ℂ⟶ℂ → 𝑝 Fn ℂ)
9997, 98syl 17 . . . . . . . . . . . . . . 15 (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
10099ad2antrr 757 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑝 Fn ℂ)
10146, 47syl 17 . . . . . . . . . . . . . . 15 (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
102101ad2antlr 758 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑎 Fn ℂ)
10349a1i 11 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
104 simplrl 795 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝𝐴) = 0)
105 simplrr 796 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
106100, 102, 103, 103, 51, 104, 105ofval 6782 . . . . . . . . . . . . 13 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
10741, 106sylan2 489 . . . . . . . . . . . 12 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
108 0m0e0 10980 . . . . . . . . . . . 12 (0 − 0) = 0
109107, 108syl6eq 2659 . . . . . . . . . . 11 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝𝑓𝑎)‘𝐴) = 0)
110109ex 448 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝𝑓𝑎)‘𝐴) = 0))
11196, 110sylan2 489 . . . . . . . . 9 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (𝐴 ∈ 𝔸 → ((𝑝𝑓𝑎)‘𝐴) = 0))
112111com12 32 . . . . . . . 8 (𝐴 ∈ 𝔸 → (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((𝑝𝑓𝑎)‘𝐴) = 0))
113112impl 647 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((𝑝𝑓𝑎)‘𝐴) = 0)
114 simpll 785 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → 𝐴 ∈ 𝔸)
115 simpl 471 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
116 simpr 475 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
11726adantl 480 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
11828adantl 480 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
119 1z 11243 . . . . . . . . . . . 12 1 ∈ ℤ
120 zq 11629 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℚ)
121 qnegcl 11640 . . . . . . . . . . . 12 (1 ∈ ℚ → -1 ∈ ℚ)
122119, 120, 121mp2b 10 . . . . . . . . . . 11 -1 ∈ ℚ
123122a1i 11 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → -1 ∈ ℚ)
124115, 116, 117, 118, 123plysub 23724 . . . . . . . . 9 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝𝑓𝑎) ∈ (Poly‘ℚ))
125124ad2antlr 758 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (𝑝𝑓𝑎) ∈ (Poly‘ℚ))
126 simplrl 795 . . . . . . . . . 10 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → 𝑝 ∈ (Poly‘ℚ))
127 simplrr 796 . . . . . . . . . 10 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → 𝑎 ∈ (Poly‘ℚ))
128 simprr1 1101 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘𝑎) = (degAA𝐴))
129 simprl1 1098 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘𝑝) = (degAA𝐴))
130128, 129eqtr4d 2646 . . . . . . . . . 10 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘𝑎) = (deg‘𝑝))
13162ad2antrr 757 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (degAA𝐴) ∈ ℕ)
132129, 131eqeltrd 2687 . . . . . . . . . 10 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘𝑝) ∈ ℕ)
133 simprl3 1100 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑝)‘(degAA𝐴)) = 1)
134129fveq2d 6092 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑝)‘(degAA𝐴)))
135129fveq2d 6092 . . . . . . . . . . . 12 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑎)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(degAA𝐴)))
136 simprr3 1103 . . . . . . . . . . . 12 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑎)‘(degAA𝐴)) = 1)
137135, 136eqtrd 2643 . . . . . . . . . . 11 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑎)‘(deg‘𝑝)) = 1)
138133, 134, 1373eqtr4d 2653 . . . . . . . . . 10 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(deg‘𝑝)))
139 eqid 2609 . . . . . . . . . . 11 (deg‘𝑝) = (deg‘𝑝)
140139dgrsub2 36548 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((deg‘𝑎) = (deg‘𝑝) ∧ (deg‘𝑝) ∈ ℕ ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(deg‘𝑝)))) → (deg‘(𝑝𝑓𝑎)) < (deg‘𝑝))
141126, 127, 130, 132, 138, 140syl23anc 1324 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘(𝑝𝑓𝑎)) < (deg‘𝑝))
142141, 129breqtrd 4603 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (deg‘(𝑝𝑓𝑎)) < (degAA𝐴))
143 dgraa0p 36562 . . . . . . . 8 ((𝐴 ∈ 𝔸 ∧ (𝑝𝑓𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝𝑓𝑎)) < (degAA𝐴)) → (((𝑝𝑓𝑎)‘𝐴) = 0 ↔ (𝑝𝑓𝑎) = 0𝑝))
144114, 125, 142, 143syl3anc 1317 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (((𝑝𝑓𝑎)‘𝐴) = 0 ↔ (𝑝𝑓𝑎) = 0𝑝))
145113, 144mpbid 220 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (𝑝𝑓𝑎) = 0𝑝)
146 df-0p 23188 . . . . . 6 0𝑝 = (ℂ × {0})
147145, 146syl6eq 2659 . . . . 5 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → (𝑝𝑓𝑎) = (ℂ × {0}))
148 ofsubeq0 10867 . . . . . . . 8 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14949, 148mp3an1 1402 . . . . . . 7 ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
15097, 46, 149syl2an 492 . . . . . 6 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
151150ad2antlr 758 . . . . 5 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
152147, 151mpbid 220 . . . 4 (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1))) → 𝑝 = 𝑎)
153152ex 448 . . 3 ((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) → ((((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)) → 𝑝 = 𝑎))
154153ralrimivva 2953 . 2 (𝐴 ∈ 𝔸 → ∀𝑝 ∈ (Poly‘ℚ)∀𝑎 ∈ (Poly‘ℚ)((((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)) → 𝑝 = 𝑎))
155 fveq2 6088 . . . . 5 (𝑝 = 𝑎 → (deg‘𝑝) = (deg‘𝑎))
156155eqeq1d 2611 . . . 4 (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘𝑎) = (degAA𝐴)))
157 fveq1 6087 . . . . 5 (𝑝 = 𝑎 → (𝑝𝐴) = (𝑎𝐴))
158157eqeq1d 2611 . . . 4 (𝑝 = 𝑎 → ((𝑝𝐴) = 0 ↔ (𝑎𝐴) = 0))
159 fveq2 6088 . . . . . 6 (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
160159fveq1d 6090 . . . . 5 (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘𝑎)‘(degAA𝐴)))
161160eqeq1d 2611 . . . 4 (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA𝐴)) = 1))
162156, 158, 1613anbi123d 1390 . . 3 (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)))
163162reu4 3366 . 2 (∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ (∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ∀𝑝 ∈ (Poly‘ℚ)∀𝑎 ∈ (Poly‘ℚ)((((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)) → 𝑝 = 𝑎)))
16493, 154, 163sylanbrc 694 1 (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  ∃!wreu 2897  Vcvv 3172  cdif 3536  wss 3539  {csn 4124   class class class wbr 4577   × cxp 5026   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  𝑓 cof 6771  cc 9791  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798   < clt 9931  cmin 10118  -cneg 10119   / cdiv 10536  cn 10870  0cn0 11142  cz 11213  cq 11623  0𝑝c0p 23187  Polycply 23689  coeffccoe 23691  degcdgr 23692  𝔸caa 23818  degAAcdgraa 36553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-q 11624  df-rp 11668  df-fz 12156  df-fzo 12293  df-fl 12413  df-mod 12489  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-0p 23188  df-ply 23693  df-coe 23695  df-dgr 23696  df-aa 23819  df-dgraa 36555
This theorem is referenced by:  mpaalem  36565
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