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Theorem mpaaeu 38037
 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 11837 . . . . . 6 ℚ ⊆ ℂ
2 eldifi 3765 . . . . . . . . . 10 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
32ad2antlr 763 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4 zssq 11833 . . . . . . . . . 10 ℤ ⊆ ℚ
5 0z 11426 . . . . . . . . . 10 0 ∈ ℤ
64, 5sselii 3633 . . . . . . . . 9 0 ∈ ℚ
7 eqid 2651 . . . . . . . . . 10 (coeff‘𝑎) = (coeff‘𝑎)
87coef2 24032 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
93, 6, 8sylancl 695 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10 dgrcl 24034 . . . . . . . . 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 6400 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13 eldifsni 4353 . . . . . . . . 9 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
1413ad2antlr 763 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15 eqid 2651 . . . . . . . . . . 11 (deg‘𝑎) = (deg‘𝑎)
1615, 7dgreq0 24066 . . . . . . . . . 10 (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
1716necon3bid 2867 . . . . . . . . 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 222 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20 qreccl 11846 . . . . . . 7 ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
2112, 19, 20syl2anc 694 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22 plyconst 24007 . . . . . 6 ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
231, 21, 22sylancr 696 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24 simpl 472 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25 simpr 476 . . . . . 6 (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26 qaddcl 11842 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
2726adantl 481 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28 qmulcl 11844 . . . . . . 7 ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
2928adantl 481 . . . . . 6 ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
3024, 25, 27, 29plymul 24019 . . . . 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 694 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) ∈ (Poly‘ℚ))
327coef3 24033 . . . . . . . . 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 6400 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
3534, 19reccld 10832 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
3634, 19recne0d 10833 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37 dgrmulc 24072 . . . . . 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 1366 . . . . 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 809 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘𝑎) = (degAA𝐴))
4038, 39eqtrd 2685 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)) = (degAA𝐴))
41 aacn 24117 . . . . . . 7 (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
4241ad2antrr 762 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝐴 ∈ ℂ)
43 ovex 6718 . . . . . . . 8 (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44 fnconstg 6131 . . . . . . . 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 23999 . . . . . . . 8 (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47 ffn 6083 . . . . . . . 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 10055 . . . . . . . 8 ℂ ∈ V
5049a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
51 inidm 3855 . . . . . . 7 (ℂ ∩ ℂ) = ℂ
5243fvconst2 6510 . . . . . . . 8 (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
5352adantl 481 . . . . . . 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 818 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
5545, 48, 50, 50, 51, 53, 54ofval 6948 . . . . . 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 703 . . . . 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 10273 . . . . 5 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0) = 0)
5856, 57eqtrd 2685 . . . 4 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0)
59 coemulc 24056 . . . . . . 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 694 . . . . . 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 6231 . . . . 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 38033 . . . . . . . 8 (𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
6362ad2antrr 762 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ)
6463nnnn0d 11389 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ0)
65 fnconstg 6131 . . . . . . . 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 6083 . . . . . . . 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 11336 . . . . . . . 8 0 ∈ V
7069a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℕ0 ∈ V)
71 inidm 3855 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
7243fvconst2 6510 . . . . . . . 8 ((degAA𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
7372adantl 481 . . . . . . 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 817 . . . . . . . . 9 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA𝐴))
7574eqcomd 2657 . . . . . . . 8 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (degAA𝐴) = (deg‘𝑎))
7675fveq2d 6233 . . . . . . 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 6948 . . . . . 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 703 . . . . 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 10835 . . . . 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 2689 . . . 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 6229 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (deg‘𝑝) = (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)))
8281eqeq1d 2653 . . . . . 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 6228 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (𝑝𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴))
8483eqeq1d 2653 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((𝑝𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0))
85 fveq2 6229 . . . . . . . 8 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)))
8685fveq1d 6231 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)))
8786eqeq1d 2653 . . . . . 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 1439 . . . . 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 3340 . . . 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 1368 . . 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 38032 . . . 4 (𝐴 ∈ 𝔸 → ((degAA𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)))
9291simprd 478 . . 3 (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0))
9390, 92r19.29a 3107 . 2 (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
94 simp2 1082 . . . . . . . . . . 11 (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝𝐴) = 0)
95 simp2 1082 . . . . . . . . . . 11 (((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1) → (𝑎𝐴) = 0)
9694, 95anim12i 589 . . . . . . . . . 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 23999 . . . . . . . . . . . . . . . 16 (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
98 ffn 6083 . . . . . . . . . . . . . . . 16 (𝑝:ℂ⟶ℂ → 𝑝 Fn ℂ)
9997, 98syl 17 . . . . . . . . . . . . . . 15 (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
10099ad2antrr 762 . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 817 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝𝐴) = 0)
105 simplrr 818 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
106100, 102, 103, 103, 51, 104, 105ofval 6948 . . . . . . . . . . . . 13 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
10741, 106sylan2 490 . . . . . . . . . . . 12 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
108 0m0e0 11168 . . . . . . . . . . . 12 (0 − 0) = 0
109107, 108syl6eq 2701 . . . . . . . . . . 11 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝𝑓𝑎)‘𝐴) = 0)
110109ex 449 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝𝑓𝑎)‘𝐴) = 0))
11196, 110sylan2 490 . . . . . . . . 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 649 . . . . . . 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 805 . . . . . . . 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 472 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
116 simpr 476 . . . . . . . . . 10 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
11726adantl 481 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
11828adantl 481 . . . . . . . . . 10 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
119 1z 11445 . . . . . . . . . . . 12 1 ∈ ℤ
120 zq 11832 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℚ)
121 qnegcl 11843 . . . . . . . . . . . 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 24020 . . . . . . . . 9 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝𝑓𝑎) ∈ (Poly‘ℚ))
125124ad2antlr 763 . . . . . . . 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 817 . . . . . . . . . 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 818 . . . . . . . . . 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 1129 . . . . . . . . . . 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 1126 . . . . . . . . . . 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 2688 . . . . . . . . . 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 762 . . . . . . . . . . 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 2730 . . . . . . . . . 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 1128 . . . . . . . . . . 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 6233 . . . . . . . . . . 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 6233 . . . . . . . . . . . 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 1131 . . . . . . . . . . . 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 2685 . . . . . . . . . . 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 2695 . . . . . . . . . 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 2651 . . . . . . . . . . 11 (deg‘𝑝) = (deg‘𝑝)
140139dgrsub2 38022 . . . . . . . . . 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 1373 . . . . . . . . 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 4711 . . . . . . . 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 38036 . . . . . . . 8 ((𝐴 ∈ 𝔸 ∧ (𝑝𝑓𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝𝑓𝑎)) < (degAA𝐴)) → (((𝑝𝑓𝑎)‘𝐴) = 0 ↔ (𝑝𝑓𝑎) = 0𝑝))
144114, 125, 142, 143syl3anc 1366 . . . . . . 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 222 . . . . . 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 23482 . . . . . 6 0𝑝 = (ℂ × {0})
147145, 146syl6eq 2701 . . . . 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 11055 . . . . . . . 8 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14949, 148mp3an1 1451 . . . . . . 7 ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
15097, 46, 149syl2an 493 . . . . . 6 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
151150ad2antlr 763 . . . . 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 222 . . . 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 449 . . 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 3000 . 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 6229 . . . . 5 (𝑝 = 𝑎 → (deg‘𝑝) = (deg‘𝑎))
156155eqeq1d 2653 . . . 4 (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘𝑎) = (degAA𝐴)))
157 fveq1 6228 . . . . 5 (𝑝 = 𝑎 → (𝑝𝐴) = (𝑎𝐴))
158157eqeq1d 2653 . . . 4 (𝑝 = 𝑎 → ((𝑝𝐴) = 0 ↔ (𝑎𝐴) = 0))
159 fveq2 6229 . . . . . 6 (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
160159fveq1d 6231 . . . . 5 (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘𝑎)‘(degAA𝐴)))
161160eqeq1d 2653 . . . 4 (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA𝐴)) = 1))
162156, 158, 1613anbi123d 1439 . . 3 (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)))
163162reu4 3433 . 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 699 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  {csn 4210   class class class wbr 4685   × cxp 5141   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937  ℂcc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   − cmin 10304  -cneg 10305   / cdiv 10722  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ℚcq 11826  0𝑝c0p 23481  Polycply 23985  coeffccoe 23987  degcdgr 23988  𝔸caa 24114  degAAcdgraa 38027 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-coe 23991  df-dgr 23992  df-aa 24115  df-dgraa 38029 This theorem is referenced by:  mpaalem  38039
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