Theorem mpaaeu 42496

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 12966	. . . . . 6 ⊢ ℚ ⊆ ℂ
2		eldifi 4122	. . . . . . . . . 10 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
3	2	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4		zssq 12962	. . . . . . . . . 10 ⊢ ℤ ⊆ ℚ
5		0z 12591	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
6	4, 5	sselii 3975	. . . . . . . . 9 ⊢ 0 ∈ ℚ
7		eqid 2727	. . . . . . . . . 10 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
8	7	coef2 26152	. . . . . . . . 9 ⊢ ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
9	3, 6, 8	sylancl 585	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10		dgrcl 26154	. . . . . . . . 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	ffvelcdmd 7089	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13		eldifsni 4789	. . . . . . . . 9 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
14	13	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15		eqid 2727	. . . . . . . . . . 11 ⊢ (deg‘𝑎) = (deg‘𝑎)
16	15, 7	dgreq0 26187	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
17	16	necon3bid 2980	. . . . . . . . 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 231	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20		qreccl 12975	. . . . . . 7 ⊢ ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
21	12, 19, 20	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22		plyconst 26127	. . . . . 6 ⊢ ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
23	1, 21, 22	sylancr 586	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
24		simpl 482	. . . . . 6 ⊢ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
25		simpr 484	. . . . . 6 ⊢ (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
26		qaddcl 12971	. . . . . . 7 ⊢ ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 + 𝑐) ∈ ℚ)
27	26	adantl 481	. . . . . 6 ⊢ ((((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
28		qmulcl 12973	. . . . . . 7 ⊢ ((𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ) → (𝑏 · 𝑐) ∈ ℚ)
29	28	adantl 481	. . . . . 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 26139	. . . . 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 583	. . . 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 26153	. . . . . . . . 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	ffvelcdmd 7089	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
35	34, 19	reccld 12005	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
36	34, 19	recne0d 12006	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37		dgrmulc 26193	. . . . . 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 1369	. . . . 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 770	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (deg‘𝑎) = (degAA‘𝐴))
40	38, 39	eqtrd 2767	. . . 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 26239	. . . . . . 7 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
42	41	ad2antrr 725	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝐴 ∈ ℂ)
43		ovex 7447	. . . . . . . 8 ⊢ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44		fnconstg 6779	. . . . . . . 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 26119	. . . . . . . 8 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47		ffn 6716	. . . . . . . 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 11211	. . . . . . . 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 4214	. . . . . . 7 ⊢ (ℂ ∩ ℂ) = ℂ
52	43	fvconst2 7210	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘𝐴) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
53	52	adantl 481	. . . . . . 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 777	. . . . . . 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 7690	. . . . . 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 686	. . . . 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 11435	. . . . 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 2767	. . . 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 26176	. . . . . . 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 583	. . . . . 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 6893	. . . . 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 42492	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
63	62	ad2antrr 725	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ)
64	63	nnnn0d 12554	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ0)
65		fnconstg 6779	. . . . . . . 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 6717	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68		nn0ex 12500	. . . . . . . 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 4214	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
71	43	fvconst2 7210	. . . . . . . 8 ⊢ ((degAA‘𝐴) ∈ ℕ0 → ((ℕ0 × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))})‘(degAA‘𝐴)) = (1 / ((coeff‘𝑎)‘(deg‘𝑎))))
72	71	adantl 481	. . . . . . 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 776	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (deg‘𝑎) = (degAA‘𝐴))
74	73	eqcomd 2733	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (degAA‘𝐴) = (deg‘𝑎))
75	74	fveq2d 6895	. . . . . . 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 7690	. . . . . 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 686	. . . . 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 12008	. . . . 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 2771	. . . 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 6900	. . . . . 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 6890	. . . . . . 7 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (𝑝‘𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴))
82	81	eqeq1d 2729	. . . . . 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 6891	. . . . . . . 8 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)))
84	83	fveq1d 6893	. . . . . . 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 2729	. . . . . 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 1433	. . . . 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 3607	. . . 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 1370	. . 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 42491	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ((degAA‘𝐴) ∈ ℕ ∧ ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)))
90	89	simprd 495	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0))
91	88, 90	r19.29a 3157	. 2 ⊢ (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
92		simp2 1135	. . . . . . . . . . 11 ⊢ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) → (𝑝‘𝐴) = 0)
93		simp2 1135	. . . . . . . . . . 11 ⊢ (((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1) → (𝑎‘𝐴) = 0)
94	92, 93	anim12i 612	. . . . . . . . . 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 26119	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
96	95	ffnd 6717	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
97	96	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → 𝑝 Fn ℂ)
98	46	ffnd 6717	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
99	98	ad2antlr 726	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ ℂ) → (𝑝‘𝐴) = 0)
102		simplrr 777	. . . . . . . . . . . . . 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 7690	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝 ∘f − 𝑎)‘𝐴) = (0 − 0))
105		0m0e0 12354	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
106	104, 105	eqtrdi 2783	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝 ∘f − 𝑎)‘𝐴) = 0)
107	106	ex 412	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → (𝐴 ∈ 𝔸 → ((𝑝 ∘f − 𝑎)‘𝐴) = 0))
108	94, 107	sylan2 592	. . . . . . . . 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 455	. . . . . . 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 766	. . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑝 ∈ (Poly‘ℚ))
113		simpr 484	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → 𝑎 ∈ (Poly‘ℚ))
114	26	adantl 481	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 + 𝑐) ∈ ℚ)
115	28	adantl 481	. . . . . . . . . 10 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ (𝑏 ∈ ℚ ∧ 𝑐 ∈ ℚ)) → (𝑏 · 𝑐) ∈ ℚ)
116		1z 12614	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
117		zq 12960	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
118		qnegcl 12972	. . . . . . . . . . . 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 26140	. . . . . . . . 9 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℚ))
122	121	ad2antlr 726	. . . . . . . 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 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))) → 𝑝 ∈ (Poly‘ℚ))
124		simplrr 777	. . . . . . . . . 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 1219	. . . . . . . . . . 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 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))) → (deg‘𝑝) = (degAA‘𝐴))
127	125, 126	eqtr4d 2770	. . . . . . . . . 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 725	. . . . . . . . . . 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 2828	. . . . . . . . . 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 1218	. . . . . . . . . . 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 6895	. . . . . . . . . . 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 6895	. . . . . . . . . . . 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 1221	. . . . . . . . . . . 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 2767	. . . . . . . . . . 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 2777	. . . . . . . . . 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 2727	. . . . . . . . . . 11 ⊢ (deg‘𝑝) = (deg‘𝑝)
137	136	dgrsub2 42481	. . . . . . . . . 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 1375	. . . . . . . . 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 5168	. . . . . . . 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 42495	. . . . . . . 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 1369	. . . . . . 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 231	. . . . . 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 25586	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
144	142, 143	eqtrdi 2783	. . . . 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 12231	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
146	49, 145	mp3an1 1445	. . . . . . 7 ⊢ ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
147	95, 46, 146	syl2an 595	. . . . . 6 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148	147	ad2antlr 726	. . . . 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 231	. . . 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 412	. . 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 3195	. 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 6900	. . . 4 ⊢ (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA‘𝐴) ↔ (deg‘𝑎) = (degAA‘𝐴)))
153		fveq1 6890	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝‘𝐴) = (𝑎‘𝐴))
154	153	eqeq1d 2729	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝‘𝐴) = 0 ↔ (𝑎‘𝐴) = 0))
155		fveq2 6891	. . . . . 6 ⊢ (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156	155	fveq1d 6893	. . . . 5 ⊢ (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA‘𝐴)) = ((coeff‘𝑎)‘(degAA‘𝐴)))
157	156	eqeq1d 2729	. . . 4 ⊢ (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA‘𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))
158	152, 154, 157	3anbi123d 1433	. . 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 582	1 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  ∃wrex 3065  ∃!wreu 3369  Vcvv 3469   ∖ cdif 3941   ⊆ wss 3944  {csn 4624   class class class wbr 5142   × cxp 5670   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∘_f cof 7677  ℂcc 11128  0cc0 11130  1c1 11131   + caddc 11133   · cmul 11135   < clt 11270   − cmin 11466  -cneg 11467   / cdiv 11893  ℕcn 12234  ℕ₀cn0 12494  ℤcz 12580  ℚcq 12954  0_𝑝c0p 25585  Polycply 26105  coeffccoe 26107  degcdgr 26108  𝔸caa 26236  deg_AAcdgraa 42486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-rlim 15457  df-sum 15657  df-0p 25586  df-ply 26109  df-coe 26111  df-dgr 26112  df-aa 26237  df-dgraa 42488

This theorem is referenced by:  mpaalem  42498