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Theorem mpaaeu 40109
 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 12349 . . . . . 6 ℚ ⊆ ℂ
2 eldifi 4054 . . . . . . . . . 10 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
32ad2antlr 726 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4 zssq 12345 . . . . . . . . . 10 ℤ ⊆ ℚ
5 0z 11982 . . . . . . . . . 10 0 ∈ ℤ
64, 5sselii 3912 . . . . . . . . 9 0 ∈ ℚ
7 eqid 2798 . . . . . . . . . 10 (coeff‘𝑎) = (coeff‘𝑎)
87coef2 24835 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
93, 6, 8sylancl 589 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10 dgrcl 24837 . . . . . . . . 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 6829 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13 eldifsni 4683 . . . . . . . . 9 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
1413ad2antlr 726 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15 eqid 2798 . . . . . . . . . . 11 (deg‘𝑎) = (deg‘𝑎)
1615, 7dgreq0 24869 . . . . . . . . . 10 (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
1716necon3bid 3031 . . . . . . . . 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 235 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20 qreccl 12358 . . . . . . 7 ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
2112, 19, 20syl2anc 587 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22 plyconst 24810 . . . . . 6 ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
231, 21, 22sylancr 590 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24 simpl 486 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25 simpr 488 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26 qaddcl 12354 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
2726adantl 485 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28 qmulcl 12356 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
2928adantl 485 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
3024, 25, 27, 29plymul 24822 . . . . 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 587 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) ∈ (Poly‘ℚ))
327coef3 24836 . . . . . . . . 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 6829 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
3534, 19reccld 11400 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
3634, 19recne0d 11401 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37 dgrmulc 24875 . . . . . 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 1368 . . . . 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 770 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘𝑎) = (degAA𝐴))
4038, 39eqtrd 2833 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA𝐴))
41 aacn 24920 . . . . . . 7 (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
4241ad2antrr 725 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝐴 ∈ ℂ)
43 ovex 7168 . . . . . . . 8 (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44 fnconstg 6541 . . . . . . . 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 24802 . . . . . . . 8 (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47 ffn 6487 . . . . . . . 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 10609 . . . . . . . 8 ℂ ∈ V
5049a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
51 inidm 4145 . . . . . . 7 (ℂ ∩ ℂ) = ℂ
5243fvconst2 6943 . . . . . . . 8 (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
5352adantl 485 . . . . . . 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 777 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
5545, 48, 50, 50, 51, 53, 54ofval 7399 . . . . . 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 686 . . . . 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 10830 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0) = 0)
5856, 57eqtrd 2833 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0)
59 coemulc 24859 . . . . . . 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 587 . . . . . 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 6647 . . . . 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 40105 . . . . . . . 8 (𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
6362ad2antrr 725 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ)
6463nnnn0d 11945 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ0)
65 fnconstg 6541 . . . . . . . 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 6488 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68 nn0ex 11893 . . . . . . . 8 0 ∈ V
6968a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℕ0 ∈ V)
70 inidm 4145 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
7143fvconst2 6943 . . . . . . . 8 ((degAA𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
7271adantl 485 . . . . . . 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 776 . . . . . . . . 9 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA𝐴))
7473eqcomd 2804 . . . . . . . 8 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (degAA𝐴) = (deg‘𝑎))
7574fveq2d 6649 . . . . . . 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 7399 . . . . . 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 686 . . . . 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 11403 . . . . 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 2837 . . . 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 6654 . . . . . 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 6644 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (𝑝𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴))
8281eqeq1d 2800 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((𝑝𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0))
83 fveq2 6645 . . . . . . . 8 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)))
8483fveq1d 6647 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA𝐴)))
8584eqeq1d 2800 . . . . . 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 1433 . . . . 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 3571 . . . 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 1369 . . 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 40104 . . . 4 (𝐴 ∈ 𝔸 → ((degAA𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)))
9089simprd 499 . . 3 (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0))
9188, 90r19.29a 3248 . 2 (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
92 simp2 1134 . . . . . . . . . . 11 (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝𝐴) = 0)
93 simp2 1134 . . . . . . . . . . 11 (((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1) → (𝑎𝐴) = 0)
9492, 93anim12i 615 . . . . . . . . . 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 24802 . . . . . . . . . . . . . . . 16 (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
9695ffnd 6488 . . . . . . . . . . . . . . 15 (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
9796ad2antrr 725 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑝 Fn ℂ)
9846ffnd 6488 . . . . . . . . . . . . . . 15 (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
9998ad2antlr 726 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝𝐴) = 0)
102 simplrr 777 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
10397, 99, 100, 100, 51, 101, 102ofval 7399 . . . . . . . . . . . . 13 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝f𝑎)‘𝐴) = (0 − 0))
10441, 103sylan2 595 . . . . . . . . . . . 12 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝f𝑎)‘𝐴) = (0 − 0))
105 0m0e0 11747 . . . . . . . . . . . 12 (0 − 0) = 0
106104, 105eqtrdi 2849 . . . . . . . . . . 11 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝f𝑎)‘𝐴) = 0)
107106ex 416 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝f𝑎)‘𝐴) = 0))
10894, 107sylan2 595 . . . . . . . . 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 459 . . . . . . 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 766 . . . . . . . 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 486 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
113 simpr 488 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
11426adantl 485 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
11528adantl 485 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
116 1z 12002 . . . . . . . . . . . 12 1 ∈ ℤ
117 zq 12344 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℚ)
118 qnegcl 12355 . . . . . . . . . . . 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 24823 . . . . . . . . 9 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝f𝑎) ∈ (Poly‘ℚ))
122121ad2antlr 726 . . . . . . . 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 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))) → 𝑝 ∈ (Poly‘ℚ))
124 simplrr 777 . . . . . . . . . 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 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‘𝑎) = (degAA𝐴))
126 simprl1 1215 . . . . . . . . . . 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 2836 . . . . . . . . . 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 725 . . . . . . . . . . 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 2890 . . . . . . . . . 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 1217 . . . . . . . . . . 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 6649 . . . . . . . . . . 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 6649 . . . . . . . . . . . 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 1220 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 2843 . . . . . . . . . 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 2798 . . . . . . . . . . 11 (deg‘𝑝) = (deg‘𝑝)
137136dgrsub2 40094 . . . . . . . . . 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 1374 . . . . . . . . 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 5056 . . . . . . . 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 40108 . . . . . . . 8 ((𝐴 ∈ 𝔸 ∧ (𝑝f𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝f𝑎)) < (degAA𝐴)) → (((𝑝f𝑎)‘𝐴) = 0 ↔ (𝑝f𝑎) = 0𝑝))
141111, 122, 139, 140syl3anc 1368 . . . . . . 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 235 . . . . . 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 24281 . . . . . 6 0𝑝 = (ℂ × {0})
144142, 143eqtrdi 2849 . . . . 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 11624 . . . . . . . 8 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14649, 145mp3an1 1445 . . . . . . 7 ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14795, 46, 146syl2an 598 . . . . . 6 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148147ad2antlr 726 . . . . 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 235 . . . 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 416 . . 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 3156 . 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 6654 . . . 4 (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘𝑎) = (degAA𝐴)))
153 fveq1 6644 . . . . 5 (𝑝 = 𝑎 → (𝑝𝐴) = (𝑎𝐴))
154153eqeq1d 2800 . . . 4 (𝑝 = 𝑎 → ((𝑝𝐴) = 0 ↔ (𝑎𝐴) = 0))
155 fveq2 6645 . . . . . 6 (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156155fveq1d 6647 . . . . 5 (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘𝑎)‘(degAA𝐴)))
157156eqeq1d 2800 . . . 4 (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA𝐴)) = 1))
158152, 154, 1573anbi123d 1433 . . 3 (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)))
159158reu4 3670 . 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 586 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  {csn 4525   class class class wbr 5030   × cxp 5517   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘f cof 7388  ℂcc 10526  0cc0 10528  1c1 10529   + caddc 10531   · cmul 10533   < clt 10666   − cmin 10861  -cneg 10862   / cdiv 11288  ℕcn 11627  ℕ0cn0 11887  ℤcz 11971  ℚcq 12338  0𝑝c0p 24280  Polycply 24788  coeffccoe 24790  degcdgr 24791  𝔸caa 24917  degAAcdgraa 40099 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606  ax-addf 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-pm 8394  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-inf 8893  df-oi 8960  df-card 9354  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-q 12339  df-rp 12380  df-fz 12888  df-fzo 13031  df-fl 13159  df-mod 13235  df-seq 13367  df-exp 13428  df-hash 13689  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-0p 24281  df-ply 24792  df-coe 24794  df-dgr 24795  df-aa 24918  df-dgraa 40101 This theorem is referenced by:  mpaalem  40111
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