Theorem mpaaeu 39757

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) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))

Distinct variable group:   𝐴,𝑝

Proof of Theorem mpaaeu

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qsscn 12362	. . . . . 6 ⊢ ℚ ⊆ ℂ
2		eldifi 4105	. . . . . . . . . 10 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
3	2	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4		zssq 12358	. . . . . . . . . 10 ⊢ ℤ ⊆ ℚ
5		0z 11995	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
6	4, 5	sselii 3966	. . . . . . . . 9 ⊢ 0 ∈ ℚ
7		eqid 2823	. . . . . . . . . 10 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
8	7	coef2 24823	. . . . . . . . 9 ⊢ ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
9	3, 6, 8	sylancl 588	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10		dgrcl 24825	. . . . . . . . 9 ⊢ (𝑎 ∈ (Poly‘ℚ) → (deg‘𝑎) ∈ ℕ0)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (deg‘𝑎) ∈ ℕ0)
12	9, 11	ffvelrnd 6854	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13		eldifsni 4724	. . . . . . . . 9 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
14	13	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15		eqid 2823	. . . . . . . . . . 11 ⊢ (deg‘𝑎) = (deg‘𝑎)
16	15, 7	dgreq0 24857	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
17	16	necon3bid 3062	. . . . . . . . 9 ⊢ (𝑎 ∈ (Poly‘ℚ) → (𝑎 ≠ 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0))
18	3, 17	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (𝑎 ≠ 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0))
19	14, 18	mpbid 234	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20		qreccl 12371	. . . . . . 7 ⊢ ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
21	12, 19, 20	syl2anc 586	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22		plyconst 24798	. . . . . 6 ⊢ ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
23	1, 21, 22	sylancr 589	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24		simpl 485	. . . . . 6 ⊢ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25		simpr 487	. . . . . 6 ⊢ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26		qaddcl 12367	. . . . . . 7 ⊢ ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
27	26	adantl 484	. . . . . 6 ⊢ ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28		qmulcl 12369	. . . . . . 7 ⊢ ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
29	28	adantl 484	. . . . . 6 ⊢ ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
30	24, 25, 27, 29	plymul 24810	. . . . 5 ⊢ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) ∈ (Poly‘ℚ))
31	23, 3, 30	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) ∈ (Poly‘ℚ))
32	7	coef3 24824	. . . . . . . . 9 ⊢ (𝑎 ∈ (Poly‘ℚ) → (coeff‘𝑎):ℕ0⟶ℂ)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℂ)
34	33, 11	ffvelrnd 6854	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
35	34, 19	reccld 11411	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
36	34, 19	recne0d 11412	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37		dgrmulc 24863	. . . . . 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))
38	35, 36, 3, 37	syl3anc 1367	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (deg‘𝑎))
39		simprl 769	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (deg‘𝑎) = (degAA‘𝐴))
40	38, 39	eqtrd 2858	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA‘𝐴))
41		aacn 24908	. . . . . . 7 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
42	41	ad2antrr 724	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝐴 ∈ ℂ)
43		ovex 7191	. . . . . . . 8 ⊢ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44		fnconstg 6569	. . . . . . . 8 ⊢ ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℂ)
45	43, 44	mp1i 13	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℂ)
46		plyf 24790	. . . . . . . 8 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47		ffn 6516	. . . . . . . 8 ⊢ (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
48	3, 46, 47	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 Fn ℂ)
49		cnex 10620	. . . . . . . 8 ⊢ ℂ ∈ V
50	49	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ℂ ∈ V)
51		inidm 4197	. . . . . . 7 ⊢ (ℂ ∩ ℂ) = ℂ
52	43	fvconst2 6968	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
53	52	adantl 484	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 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) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎‘𝐴) = 0)
55	45, 48, 50, 50, 51, 53, 54	ofval 7420	. . . . . 6 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0))
56	42, 55	mpdan 685	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0))
57	35	mul01d 10841	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · 0) = 0)
58	56, 57	eqtrd 2858	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0)
59		coemulc 24847	. . . . . . 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)))
60	35, 3, 59	syl2anc 586	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · (coeff‘𝑎)))
61	60	fveq1d 6674	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)) = (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · (coeff‘𝑎))‘(degAA‘𝐴)))
62		dgraacl 39753	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
63	62	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ)
64	63	nnnn0d 11958	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ0)
65		fnconstg 6569	. . . . . . . 8 ⊢ ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V → (ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℕ0)
66	43, 65	mp1i 13	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) Fn ℕ0)
67	33	ffnd 6517	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68		nn0ex 11906	. . . . . . . 8 ⊢ ℕ0 ∈ V
69	68	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ℕ0 ∈ V)
70		inidm 4197	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
71	43	fvconst2 6968	. . . . . . . 8 ⊢ ((degAA‘𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA‘𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
72	71	adantl 484	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA‘𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
73		simplrl 775	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA‘𝐴))
74	73	eqcomd 2829	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (degAA‘𝐴) = (deg‘𝑎))
75	74	fveq2d 6676	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → ((coeff‘𝑎)‘(degAA‘𝐴)) = ((coeff‘𝑎)‘(deg‘𝑎)))
76	66, 67, 69, 69, 70, 72, 75	ofval 7420	. . . . . 6 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · (coeff‘𝑎))‘(degAA‘𝐴)) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))))
77	64, 76	mpdan 685	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · (coeff‘𝑎))‘(degAA‘𝐴)) = ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))))
78	34, 19	recid2d 11414	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((1 / ((coeff‘𝑎)‘(deg‘𝑎))) · ((coeff‘𝑎)‘(deg‘𝑎))) = 1)
79	61, 77, 78	3eqtrd 2862	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)) = 1)
80		fveqeq2 6681	. . . . . 6 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((deg‘𝑝) = (degAA‘𝐴) ↔ (deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA‘𝐴)))
81		fveq1 6671	. . . . . . 7 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (𝑝‘𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴))
82	81	eqeq1d 2825	. . . . . 6 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((𝑝‘𝐴) = 0 ↔ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0))
83		fveq2 6672	. . . . . . . 8 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)))
84	83	fveq1d 6674	. . . . . . 7 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → ((coeff‘𝑝)‘(degAA‘𝐴)) = ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)))
85	84	eqeq1d 2825	. . . . . 6 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (((coeff‘𝑝)‘(degAA‘𝐴)) = 1 ↔ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)) = 1))
86	80, 82, 85	3anbi123d 1432	. . . . 5 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ↔ ((deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA‘𝐴) ∧ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0 ∧ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)) = 1)))
87	86	rspcev 3625	. . . 4 ⊢ ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) ∈ (Poly‘ℚ) ∧ ((deg‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)) = (degAA‘𝐴) ∧ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴) = 0 ∧ ((coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎))‘(degAA‘𝐴)) = 1)) → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
88	31, 40, 58, 79, 87	syl13anc 1368	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
89		dgraalem 39752	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ((degAA‘𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)))
90	89	simprd 498	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0))
91	88, 90	r19.29a 3291	. 2 ⊢ (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
92		simp2 1133	. . . . . . . . . . 11 ⊢ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) → (𝑝‘𝐴) = 0)
93		simp2 1133	. . . . . . . . . . 11 ⊢ (((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1) → (𝑎‘𝐴) = 0)
94	92, 93	anim12i 614	. . . . . . . . . 10 ⊢ ((((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)) → ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0))
95		plyf 24790	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
96	95	ffnd 6517	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
97	96	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → 𝑝 Fn ℂ)
98	46	ffnd 6517	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
99	98	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → 𝑎 Fn ℂ)
100	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → ℂ ∈ V)
101		simplrl 775	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝‘𝐴) = 0)
102		simplrr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑎‘𝐴) = 0)
103	97, 99, 100, 100, 51, 101, 102	ofval 7420	. . . . . . . . . . . . 13 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → ((𝑝 ∘f − 𝑎)‘𝐴) = (0 − 0))
104	41, 103	sylan2 594	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝 ∘f − 𝑎)‘𝐴) = (0 − 0))
105		0m0e0 11760	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
106	104, 105	syl6eq 2874	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝 ∘f − 𝑎)‘𝐴) = 0)
107	106	ex 415	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝 ∘f − 𝑎)‘𝐴) = 0))
108	94, 107	sylan2 594	. . . . . . . . 9 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (𝐴 ∈ 𝔸 → ((𝑝 ∘f − 𝑎)‘𝐴) = 0))
109	108	com12 32	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((𝑝 ∘f − 𝑎)‘𝐴) = 0))
110	109	impl 458	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((𝑝 ∘f − 𝑎)‘𝐴) = 0)
111		simpll 765	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → 𝐴 ∈ 𝔸)
112		simpl 485	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
113		simpr 487	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
114	26	adantl 484	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
115	28	adantl 484	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
116		1z 12015	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
117		zq 12357	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
118		qnegcl 12368	. . . . . . . . . . . 12 ⊢ (1 ∈ ℚ → -1 ∈ ℚ)
119	116, 117, 118	mp2b 10	. . . . . . . . . . 11 ⊢ -1 ∈ ℚ
120	119	a1i 11	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → -1 ∈ ℚ)
121	112, 113, 114, 115, 120	plysub 24811	. . . . . . . . 9 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℚ))
122	121	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℚ))
123		simplrl 775	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → 𝑝 ∈ (Poly‘ℚ))
124		simplrr 776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → 𝑎 ∈ (Poly‘ℚ))
125		simprr1 1217	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘𝑎) = (degAA‘𝐴))
126		simprl1 1214	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘𝑝) = (degAA‘𝐴))
127	125, 126	eqtr4d 2861	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘𝑎) = (deg‘𝑝))
128	62	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (degAA‘𝐴) ∈ ℕ)
129	126, 128	eqeltrd 2915	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘𝑝) ∈ ℕ)
130		simprl3 1216	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)
131	126	fveq2d 6676	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑝)‘(degAA‘𝐴)))
132	126	fveq2d 6676	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑎)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(degAA‘𝐴)))
133		simprr3 1219	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)
134	132, 133	eqtrd 2858	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑎)‘(deg‘𝑝)) = 1)
135	130, 131, 134	3eqtr4d 2868	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(deg‘𝑝)))
136		eqid 2823	. . . . . . . . . . 11 ⊢ (deg‘𝑝) = (deg‘𝑝)
137	136	dgrsub2 39742	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((deg‘𝑎) = (deg‘𝑝) ∧ (deg‘𝑝) ∈ ℕ ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑎)‘(deg‘𝑝)))) → (deg‘(𝑝 ∘f − 𝑎)) < (deg‘𝑝))
138	123, 124, 127, 129, 135, 137	syl23anc 1373	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘(𝑝 ∘f − 𝑎)) < (deg‘𝑝))
139	138, 126	breqtrd 5094	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (deg‘(𝑝 ∘f − 𝑎)) < (degAA‘𝐴))
140		dgraa0p 39756	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝔸 ∧ (𝑝 ∘f − 𝑎) ∈ (Poly‘ℚ) ∧ (deg‘(𝑝 ∘f − 𝑎)) < (degAA‘𝐴)) → (((𝑝 ∘f − 𝑎)‘𝐴) = 0 ↔ (𝑝 ∘f − 𝑎) = 0𝑝))
141	111, 122, 139, 140	syl3anc 1367	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (((𝑝 ∘f − 𝑎)‘𝐴) = 0 ↔ (𝑝 ∘f − 𝑎) = 0𝑝))
142	110, 141	mpbid 234	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (𝑝 ∘f − 𝑎) = 0𝑝)
143		df-0p 24273	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
144	142, 143	syl6eq 2874	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → (𝑝 ∘f − 𝑎) = (ℂ × {0}))
145		ofsubeq0 11637	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
146	49, 145	mp3an1 1444	. . . . . . 7 ⊢ ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
147	95, 46, 146	syl2an 597	. . . . . 6 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148	147	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
149	144, 148	mpbid 234	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) ∧ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))) → 𝑝 = 𝑎)
150	149	ex 415	. . 3 ⊢ ((𝐴 ∈ 𝔸 ∧ (𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ))) → ((((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)) → 𝑝 = 𝑎))
151	150	ralrimivva 3193	. 2 ⊢ (𝐴 ∈ 𝔸 → ∀𝑝 ∈ (Poly‘ℚ)∀𝑎 ∈ (Poly‘ℚ)((((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)) → 𝑝 = 𝑎))
152		fveqeq2 6681	. . . 4 ⊢ (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA‘𝐴) ↔ (deg‘𝑎) = (degAA‘𝐴)))
153		fveq1 6671	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝‘𝐴) = (𝑎‘𝐴))
154	153	eqeq1d 2825	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝‘𝐴) = 0 ↔ (𝑎‘𝐴) = 0))
155		fveq2 6672	. . . . . 6 ⊢ (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156	155	fveq1d 6674	. . . . 5 ⊢ (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA‘𝐴)) = ((coeff‘𝑎)‘(degAA‘𝐴)))
157	156	eqeq1d 2825	. . . 4 ⊢ (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA‘𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))
158	152, 154, 157	3anbi123d 1432	. . 3 ⊢ (𝑝 = 𝑎 → (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ↔ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)))
159	158	reu4 3724	. 2 ⊢ (∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ↔ (∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ∀𝑝 ∈ (Poly‘ℚ)∀𝑎 ∈ (Poly‘ℚ)((((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1)) → 𝑝 = 𝑎)))
160	91, 151, 159	sylanbrc 585	1 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  ∃!wreu 3142  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  {csn 4569   class class class wbr 5068   × cxp 5555   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409  ℂcc 10537  0cc0 10539  1c1 10540   + caddc 10542   · cmul 10544   < clt 10677   − cmin 10872  -cneg 10873   / cdiv 11299  ℕcn 11640  ℕ₀cn0 11900  ℤcz 11984  ℚcq 12351  0_𝑝c0p 24272  Polycply 24776  coeffccoe 24778  degcdgr 24779  𝔸caa 24905  deg_AAcdgraa 39747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-rlim 14848  df-sum 15045  df-0p 24273  df-ply 24780  df-coe 24782  df-dgr 24783  df-aa 24906  df-dgraa 39749

This theorem is referenced by:  mpaalem  39759