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Theorem mpaaeu 41463
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 12885 . . . . . 6 ℚ ⊆ ℂ
2 eldifi 4086 . . . . . . . . . 10 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
32ad2antlr 725 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4 zssq 12881 . . . . . . . . . 10 ℤ ⊆ ℚ
5 0z 12510 . . . . . . . . . 10 0 ∈ ℤ
64, 5sselii 3941 . . . . . . . . 9 0 ∈ ℚ
7 eqid 2736 . . . . . . . . . 10 (coeff‘𝑎) = (coeff‘𝑎)
87coef2 25592 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
93, 6, 8sylancl 586 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10 dgrcl 25594 . . . . . . . . 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, 11ffvelcdmd 7036 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13 eldifsni 4750 . . . . . . . . 9 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
1413ad2antlr 725 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15 eqid 2736 . . . . . . . . . . 11 (deg‘𝑎) = (deg‘𝑎)
1615, 7dgreq0 25626 . . . . . . . . . 10 (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
1716necon3bid 2988 . . . . . . . . 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 231 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20 qreccl 12894 . . . . . . 7 ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
2112, 19, 20syl2anc 584 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22 plyconst 25567 . . . . . 6 ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
231, 21, 22sylancr 587 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24 simpl 483 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25 simpr 485 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26 qaddcl 12890 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
2726adantl 482 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28 qmulcl 12892 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
2928adantl 482 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
3024, 25, 27, 29plymul 25579 . . . . 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 584 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) ∈ (Poly‘ℚ))
327coef3 25593 . . . . . . . . 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, 11ffvelcdmd 7036 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
3534, 19reccld 11924 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
3634, 19recne0d 11925 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37 dgrmulc 25632 . . . . . 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 1371 . . . . 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 769 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘𝑎) = (degAA𝐴))
4038, 39eqtrd 2776 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA𝐴))
41 aacn 25677 . . . . . . 7 (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
4241ad2antrr 724 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝐴 ∈ ℂ)
43 ovex 7390 . . . . . . . 8 (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44 fnconstg 6730 . . . . . . . 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 25559 . . . . . . . 8 (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47 ffn 6668 . . . . . . . 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 11132 . . . . . . . 8 ℂ ∈ V
5049a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
51 inidm 4178 . . . . . . 7 (ℂ ∩ ℂ) = ℂ
5243fvconst2 7153 . . . . . . . 8 (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
5352adantl 482 . . . . . . 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 776 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
5545, 48, 50, 50, 51, 53, 54ofval 7628 . . . . . 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 685 . . . . 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 11354 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0) = 0)
5856, 57eqtrd 2776 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0)
59 coemulc 25616 . . . . . . 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 584 . . . . . 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 6844 . . . . 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 41459 . . . . . . . 8 (𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
6362ad2antrr 724 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ)
6463nnnn0d 12473 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ0)
65 fnconstg 6730 . . . . . . . 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)
6733ffnd 6669 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68 nn0ex 12419 . . . . . . . 8 0 ∈ V
6968a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℕ0 ∈ V)
70 inidm 4178 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
7143fvconst2 7153 . . . . . . . 8 ((degAA𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
7271adantl 482 . . . . . . 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))))
73 simplrl 775 . . . . . . . . 9 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA𝐴))
7473eqcomd 2742 . . . . . . . 8 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (degAA𝐴) = (deg‘𝑎))
7574fveq2d 6846 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → ((coeff‘𝑎)‘(degAA𝐴)) = ((coeff‘𝑎)‘(deg‘𝑎)))
7666, 67, 69, 69, 70, 72, 75ofval 7628 . . . . . 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))))
7764, 76mpdan 685 . . . . 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))))
7834, 19recid2d 11927 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))) = 1)
7961, 77, 783eqtrd 2780 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA𝐴)) = 1)
80 fveqeq2 6851 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA𝐴)))
81 fveq1 6841 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (𝑝𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴))
8281eqeq1d 2738 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((𝑝𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0))
83 fveq2 6842 . . . . . . . 8 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)))
8483fveq1d 6844 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA𝐴)))
8584eqeq1d 2738 . . . . . 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))
8680, 82, 853anbi123d 1436 . . . . 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)))
8786rspcev 3581 . . . 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))
8831, 40, 58, 79, 87syl13anc 1372 . . 3 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
89 dgraalem 41458 . . . 4 (𝐴 ∈ 𝔸 → ((degAA𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)))
9089simprd 496 . . 3 (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0))
9188, 90r19.29a 3159 . 2 (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
92 simp2 1137 . . . . . . . . . . 11 (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝𝐴) = 0)
93 simp2 1137 . . . . . . . . . . 11 (((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1) → (𝑎𝐴) = 0)
9492, 93anim12i 613 . . . . . . . . . 10 ((((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)) → ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0))
95 plyf 25559 . . . . . . . . . . . . . . . 16 (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
9695ffnd 6669 . . . . . . . . . . . . . . 15 (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
9796ad2antrr 724 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑝 Fn ℂ)
9846ffnd 6669 . . . . . . . . . . . . . . 15 (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
9998ad2antlr 725 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑎 Fn ℂ)
10049a1i 11 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
101 simplrl 775 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝𝐴) = 0)
102 simplrr 776 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
10397, 99, 100, 100, 51, 101, 102ofval 7628 . . . . . . . . . . . . 13 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝f𝑎)‘𝐴) = (0 − 0))
10441, 103sylan2 593 . . . . . . . . . . . 12 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝f𝑎)‘𝐴) = (0 − 0))
105 0m0e0 12273 . . . . . . . . . . . 12 (0 − 0) = 0
106104, 105eqtrdi 2792 . . . . . . . . . . 11 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝f𝑎)‘𝐴) = 0)
107106ex 413 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝f𝑎)‘𝐴) = 0))
10894, 107sylan2 593 . . . . . . . . 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))
109108com12 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))
110109impl 456 . . . . . . 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)
111 simpll 765 . . . . . . . 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)
112 simpl 483 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
113 simpr 485 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
11426adantl 482 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
11528adantl 482 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
116 1z 12533 . . . . . . . . . . . 12 1 ∈ ℤ
117 zq 12879 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℚ)
118 qnegcl 12891 . . . . . . . . . . . 12 (1 ∈ ℚ → -1 ∈ ℚ)
119116, 117, 118mp2b 10 . . . . . . . . . . 11 -1 ∈ ℚ
120119a1i 11 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → -1 ∈ ℚ)
121112, 113, 114, 115, 120plysub 25580 . . . . . . . . 9 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝f𝑎) ∈ (Poly‘ℚ))
122121ad2antlr 725 . . . . . . . 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))
123 simplrl 775 . . . . . . . . . 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))
124 simplrr 776 . . . . . . . . . 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))
125 simprr1 1221 . . . . . . . . . . 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))
126 simprl1 1218 . . . . . . . . . . 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))
127125, 126eqtr4d 2779 . . . . . . . . . 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))
12862ad2antrr 724 . . . . . . . . . . 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)
129126, 128eqeltrd 2838 . . . . . . . . . 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)
130 simprl3 1220 . . . . . . . . . . 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)
131126fveq2d 6846 . . . . . . . . . . 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)))
132126fveq2d 6846 . . . . . . . . . . . 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)))
133 simprr3 1223 . . . . . . . . . . . 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)
134132, 133eqtrd 2776 . . . . . . . . . . 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)
135130, 131, 1343eqtr4d 2786 . . . . . . . . . 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)))
136 eqid 2736 . . . . . . . . . . 11 (deg‘𝑝) = (deg‘𝑝)
137136dgrsub2 41448 . . . . . . . . . 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))
138123, 124, 127, 129, 135, 137syl23anc 1377 . . . . . . . . 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))
139138, 126breqtrd 5131 . . . . . . . 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))
140 dgraa0p 41462 . . . . . . . 8 ((𝐴 ∈ 𝔸 ∧ (𝑝f𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝f𝑎)) < (degAA𝐴)) → (((𝑝f𝑎)‘𝐴) = 0 ↔ (𝑝f𝑎) = 0𝑝))
141111, 122, 139, 140syl3anc 1371 . . . . . . 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))
142110, 141mpbid 231 . . . . . 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)
143 df-0p 25034 . . . . . 6 0𝑝 = (ℂ × {0})
144142, 143eqtrdi 2792 . . . . 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}))
145 ofsubeq0 12150 . . . . . . . 8 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14649, 145mp3an1 1448 . . . . . . 7 ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14795, 46, 146syl2an 596 . . . . . 6 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148147ad2antlr 725 . . . . 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))
149144, 148mpbid 231 . . . 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)
150149ex 413 . . 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))
151150ralrimivva 3197 . 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))
152 fveqeq2 6851 . . . 4 (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘𝑎) = (degAA𝐴)))
153 fveq1 6841 . . . . 5 (𝑝 = 𝑎 → (𝑝𝐴) = (𝑎𝐴))
154153eqeq1d 2738 . . . 4 (𝑝 = 𝑎 → ((𝑝𝐴) = 0 ↔ (𝑎𝐴) = 0))
155 fveq2 6842 . . . . . 6 (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156155fveq1d 6844 . . . . 5 (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘𝑎)‘(degAA𝐴)))
157156eqeq1d 2738 . . . 4 (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA𝐴)) = 1))
158152, 154, 1573anbi123d 1436 . . 3 (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)))
159158reu4 3689 . 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)))
16091, 151, 159sylanbrc 583 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  Vcvv 3445  cdif 3907  wss 3910  {csn 4586   class class class wbr 5105   × cxp 5631   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cmin 11385  -cneg 11386   / cdiv 11812  cn 12153  0cn0 12413  cz 12499  cq 12873  0𝑝c0p 25033  Polycply 25545  coeffccoe 25547  degcdgr 25548  𝔸caa 25674  degAAcdgraa 41453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-coe 25551  df-dgr 25552  df-aa 25675  df-dgraa 41455
This theorem is referenced by:  mpaalem  41465
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