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Theorem mpaaeu 39256
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 12213 . . . . . 6 ℚ ⊆ ℂ
2 eldifi 4030 . . . . . . . . . 10 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
32ad2antlr 723 . . . . . . . . 9 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4 zssq 12209 . . . . . . . . . 10 ℤ ⊆ ℚ
5 0z 11846 . . . . . . . . . 10 0 ∈ ℤ
64, 5sselii 3892 . . . . . . . . 9 0 ∈ ℚ
7 eqid 2797 . . . . . . . . . 10 (coeff‘𝑎) = (coeff‘𝑎)
87coef2 24508 . . . . . . . . 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 24510 . . . . . . . . 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 6724 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13 eldifsni 4635 . . . . . . . . 9 (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
1413ad2antlr 723 . . . . . . . 8 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15 eqid 2797 . . . . . . . . . . 11 (deg‘𝑎) = (deg‘𝑎)
1615, 7dgreq0 24542 . . . . . . . . . 10 (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
1716necon3bid 3030 . . . . . . . . 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 233 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20 qreccl 12222 . . . . . . 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 24483 . . . . . 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 12218 . . . . . . 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 12220 . . . . . . 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 24495 . . . . 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 24509 . . . . . . . . 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 6724 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
3534, 19reccld 11263 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
3634, 19recne0d 11264 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37 dgrmulc 24548 . . . . . 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 1364 . . . . 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 767 . . . . 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 24593 . . . . . . 7 (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
4241ad2antrr 722 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → 𝐴 ∈ ℂ)
43 ovex 7055 . . . . . . . 8 (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44 fnconstg 6442 . . . . . . . 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 24475 . . . . . . . 8 (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47 ffn 6389 . . . . . . . 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 10471 . . . . . . . 8 ℂ ∈ V
5049a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℂ ∈ V)
51 inidm 4121 . . . . . . 7 (ℂ ∩ ℂ) = ℂ
5243fvconst2 6840 . . . . . . . 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 774 . . . . . . 7 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
5545, 48, 50, 50, 51, 53, 54ofval 7283 . . . . . 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 683 . . . . 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 10692 . . . . 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 24532 . . . . . . 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 6547 . . . . 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 39252 . . . . . . . 8 (𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
6362ad2antrr 722 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ)
6463nnnn0d 11809 . . . . . 6 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (degAA𝐴) ∈ ℕ0)
65 fnconstg 6442 . . . . . . . 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 6390 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68 nn0ex 11757 . . . . . . . 8 0 ∈ V
6968a1i 11 . . . . . . 7 (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) → ℕ0 ∈ V)
70 inidm 4121 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
7143fvconst2 6840 . . . . . . . 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 773 . . . . . . . . 9 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA𝐴))
7473eqcomd 2803 . . . . . . . 8 ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0)) ∧ (degAA𝐴) ∈ ℕ0) → (degAA𝐴) = (deg‘𝑎))
7574fveq2d 6549 . . . . . . 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 7283 . . . . . 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 683 . . . . 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 11266 . . . . 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 6554 . . . . . 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 6544 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (𝑝𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴))
8281eqeq1d 2799 . . . . . 6 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((𝑝𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)‘𝐴) = 0))
83 fveq2 6545 . . . . . . . 8 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎)))
8483fveq1d 6547 . . . . . . 7 (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘𝑓 · 𝑎))‘(degAA𝐴)))
8584eqeq1d 2799 . . . . . 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 1428 . . . . 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 3561 . . . 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 1365 . . 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 39251 . . . 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 3254 . 2 (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
92 simp2 1130 . . . . . . . . . . 11 (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝𝐴) = 0)
93 simp2 1130 . . . . . . . . . . 11 (((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1) → (𝑎𝐴) = 0)
9492, 93anim12i 612 . . . . . . . . . 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 24475 . . . . . . . . . . . . . . . 16 (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
9695ffnd 6390 . . . . . . . . . . . . . . 15 (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
9796ad2antrr 722 . . . . . . . . . . . . . 14 (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) → 𝑝 Fn ℂ)
9846ffnd 6390 . . . . . . . . . . . . . . 15 (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
9998ad2antlr 723 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝𝐴) = 0)
102 simplrr 774 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎𝐴) = 0)
10397, 99, 100, 100, 51, 101, 102ofval 7283 . . . . . . . . . . . . 13 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
10441, 103sylan2 592 . . . . . . . . . . . 12 ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝𝐴) = 0 ∧ (𝑎𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝𝑓𝑎)‘𝐴) = (0 − 0))
105 0m0e0 11611 . . . . . . . . . . . 12 (0 − 0) = 0
106104, 105syl6eq 2849 . . . . . . . . . . 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 592 . . . . . . . . 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 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))) → 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 11866 . . . . . . . . . . . 12 1 ∈ ℤ
117 zq 12207 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℚ)
118 qnegcl 12219 . . . . . . . . . . . 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 24496 . . . . . . . . 9 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝𝑓𝑎) ∈ (Poly‘ℚ))
122121ad2antlr 723 . . . . . . . 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 773 . . . . . . . . . 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 774 . . . . . . . . . 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 1214 . . . . . . . . . . 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 1211 . . . . . . . . . . 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 722 . . . . . . . . . . 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 2885 . . . . . . . . . 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 1213 . . . . . . . . . . 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 6549 . . . . . . . . . . 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 6549 . . . . . . . . . . . 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 1216 . . . . . . . . . . . 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 2797 . . . . . . . . . . 11 (deg‘𝑝) = (deg‘𝑝)
137136dgrsub2 39241 . . . . . . . . . 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 1370 . . . . . . . . 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 4994 . . . . . . . 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 39255 . . . . . . . 8 ((𝐴 ∈ 𝔸 ∧ (𝑝𝑓𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝𝑓𝑎)) < (degAA𝐴)) → (((𝑝𝑓𝑎)‘𝐴) = 0 ↔ (𝑝𝑓𝑎) = 0𝑝))
141111, 122, 139, 140syl3anc 1364 . . . . . . 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 233 . . . . . 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 23958 . . . . . 6 0𝑝 = (ℂ × {0})
144142, 143syl6eq 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 11489 . . . . . . . 8 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14649, 145mp3an1 1440 . . . . . . 7 ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
14795, 46, 146syl2an 595 . . . . . 6 ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148147ad2antlr 723 . . . . 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 233 . . . 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 3160 . 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 6554 . . . 4 (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘𝑎) = (degAA𝐴)))
153 fveq1 6544 . . . . 5 (𝑝 = 𝑎 → (𝑝𝐴) = (𝑎𝐴))
154153eqeq1d 2799 . . . 4 (𝑝 = 𝑎 → ((𝑝𝐴) = 0 ↔ (𝑎𝐴) = 0))
155 fveq2 6545 . . . . . 6 (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156155fveq1d 6547 . . . . 5 (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘𝑎)‘(degAA𝐴)))
157156eqeq1d 2799 . . . 4 (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA𝐴)) = 1))
158152, 154, 1573anbi123d 1428 . . 3 (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA𝐴) ∧ (𝑎𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA𝐴)) = 1)))
159158reu4 3661 . 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wral 3107  wrex 3108  ∃!wreu 3109  Vcvv 3440  cdif 3862  wss 3865  {csn 4478   class class class wbr 4968   × cxp 5448   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  𝑓 cof 7272  cc 10388  0cc0 10390  1c1 10391   + caddc 10393   · cmul 10395   < clt 10528  cmin 10723  -cneg 10724   / cdiv 11151  cn 11492  0cn0 11751  cz 11835  cq 12201  0𝑝c0p 23957  Polycply 24461  coeffccoe 24463  degcdgr 24464  𝔸caa 24590  degAAcdgraa 39246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-oi 8827  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-q 12202  df-rp 12244  df-fz 12747  df-fzo 12888  df-fl 13016  df-mod 13092  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-rlim 14684  df-sum 14881  df-0p 23958  df-ply 24465  df-coe 24467  df-dgr 24468  df-aa 24591  df-dgraa 39248
This theorem is referenced by:  mpaalem  39258
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