Theorem mpaaeu 43257

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 12868	. . . . . 6 ⊢ ℚ ⊆ ℂ
2		eldifi 4082	. . . . . . . . . 10 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ∈ (Poly‘ℚ))
3	2	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ∈ (Poly‘ℚ))
4		zssq 12864	. . . . . . . . . 10 ⊢ ℤ ⊆ ℚ
5		0z 12489	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
6	4, 5	sselii 3928	. . . . . . . . 9 ⊢ 0 ∈ ℚ
7		eqid 2733	. . . . . . . . . 10 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
8	7	coef2 26173	. . . . . . . . 9 ⊢ ((𝑎 ∈ (Poly‘ℚ) ∧ 0 ∈ ℚ) → (coeff‘𝑎):ℕ0⟶ℚ)
9	3, 6, 8	sylancl 586	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎):ℕ0⟶ℚ)
10		dgrcl 26175	. . . . . . . . 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 7027	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ)
13		eldifsni 4743	. . . . . . . . 9 ⊢ (𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝}) → 𝑎 ≠ 0𝑝)
14	13	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝑎 ≠ 0𝑝)
15		eqid 2733	. . . . . . . . . . 11 ⊢ (deg‘𝑎) = (deg‘𝑎)
16	15, 7	dgreq0 26208	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℚ) → (𝑎 = 0𝑝 ↔ ((coeff‘𝑎)‘(deg‘𝑎)) = 0))
17	16	necon3bid 2974	. . . . . . . . 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 232	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0)
20		qreccl 12877	. . . . . . 7 ⊢ ((((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℚ ∧ ((coeff‘𝑎)‘(deg‘𝑎)) ≠ 0) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
21	12, 19, 20	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ)
22		plyconst 26148	. . . . . 6 ⊢ ((ℚ ⊆ ℂ ∧ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℚ) → (ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∈ (Poly‘ℚ))
23	1, 21, 22	sylancr 587	. . . . 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 12873	. . . . . . 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 12875	. . . . . . 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 26160	. . . . 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 584	. . . 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 26174	. . . . . . . . 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 7027	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → ((coeff‘𝑎)‘(deg‘𝑎)) ∈ ℂ)
35	34, 19	reccld 11900	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ ℂ)
36	34, 19	recne0d 11901	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ≠ 0)
37		dgrmulc 26214	. . . . . 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 1373	. . . . 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 2768	. . . 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 26262	. . . . . . 7 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
42	41	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → 𝐴 ∈ ℂ)
43		ovex 7388	. . . . . . . 8 ⊢ (1 / ((coeff‘𝑎)‘(deg‘𝑎))) ∈ V
44		fnconstg 6719	. . . . . . . 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 26140	. . . . . . . 8 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎:ℂ⟶ℂ)
47		ffn 6659	. . . . . . . 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 11097	. . . . . . . 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 4178	. . . . . . 7 ⊢ (ℂ ∩ ℂ) = ℂ
52	43	fvconst2 7147	. . . . . . . 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 7630	. . . . . 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 687	. . . . 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 11322	. . . . 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 2768	. . . 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 26197	. . . . . . 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 584	. . . . . 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 6833	. . . . 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 43253	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
63	62	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ)
64	63	nnnn0d 12452	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (degAA‘𝐴) ∈ ℕ0)
65		fnconstg 6719	. . . . . . . 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 6660	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) → (coeff‘𝑎) Fn ℕ0)
68		nn0ex 12397	. . . . . . . 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 4178	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
71	43	fvconst2 7147	. . . . . . . 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 2739	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝑎 ∈ ((Poly‘ℚ) ∖ {0𝑝})) ∧ ((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0)) ∧ (degAA‘𝐴) ∈ ℕ0) → (degAA‘𝐴) = (deg‘𝑎))
75	74	fveq2d 6835	. . . . . . 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 7630	. . . . . 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 687	. . . . 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 11903	. . . . 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 2772	. . . 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 6840	. . . . . 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 6830	. . . . . . 7 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (𝑝‘𝐴) = (((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)‘𝐴))
82	81	eqeq1d 2735	. . . . . 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 6831	. . . . . . . 8 ⊢ (𝑝 = ((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎) → (coeff‘𝑝) = (coeff‘((ℂ × {(1 / ((coeff‘𝑎)‘(deg‘𝑎)))}) ∘f · 𝑎)))
84	83	fveq1d 6833	. . . . . . 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 2735	. . . . . 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 1438	. . . . 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 3574	. . . 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 1374	. . 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 43252	. . . 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 3142	. 2 ⊢ (𝐴 ∈ 𝔸 → ∃𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
92		simp2 1137	. . . . . . . . . . 11 ⊢ (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) → (𝑝‘𝐴) = 0)
93		simp2 1137	. . . . . . . . . . 11 ⊢ (((deg‘𝑎) = (degAA‘𝐴) ∧ (𝑎‘𝐴) = 0 ∧ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1) → (𝑎‘𝐴) = 0)
94	92, 93	anim12i 613	. . . . . . . . . 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 26140	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝:ℂ⟶ℂ)
96	95	ffnd 6660	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (Poly‘ℚ) → 𝑝 Fn ℂ)
97	96	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) → 𝑝 Fn ℂ)
98	46	ffnd 6660	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (Poly‘ℚ) → 𝑎 Fn ℂ)
99	98	ad2antlr 727	. . . . . . . . . . . . . 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 7630	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) ∧ ((𝑝‘𝐴) = 0 ∧ (𝑎‘𝐴) = 0)) ∧ 𝐴 ∈ 𝔸) → ((𝑝 ∘f − 𝑎)‘𝐴) = (0 − 0))
105		0m0e0 12250	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
106	104, 105	eqtrdi 2784	. . . . . . . . . . 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 593	. . . . . . . . 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 12512	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
117		zq 12862	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
118		qnegcl 12874	. . . . . . . . . . . 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 26161	. . . . . . . . 9 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℚ))
122	121	ad2antlr 727	. . . . . . . 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 1222	. . . . . . . . . . 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 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‘𝑝) = (degAA‘𝐴))
127	125, 126	eqtr4d 2771	. . . . . . . . . 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 726	. . . . . . . . . . 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 2833	. . . . . . . . . 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 1221	. . . . . . . . . . 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 6835	. . . . . . . . . . 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 6835	. . . . . . . . . . . 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 1224	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2778	. . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (deg‘𝑝) = (deg‘𝑝)
137	136	dgrsub2 43242	. . . . . . . . . 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 1379	. . . . . . . . 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 5121	. . . . . . . 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 43256	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 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 25608	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
144	142, 143	eqtrdi 2784	. . . . 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 12132	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
146	49, 145	mp3an1 1450	. . . . . . 7 ⊢ ((𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
147	95, 46, 146	syl2an 596	. . . . . 6 ⊢ ((𝑝 ∈ (Poly‘ℚ) ∧ 𝑎 ∈ (Poly‘ℚ)) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
148	147	ad2antlr 727	. . . . 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 232	. . . 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 3177	. 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 6840	. . . 4 ⊢ (𝑝 = 𝑎 → ((deg‘𝑝) = (degAA‘𝐴) ↔ (deg‘𝑎) = (degAA‘𝐴)))
153		fveq1 6830	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝‘𝐴) = (𝑎‘𝐴))
154	153	eqeq1d 2735	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝‘𝐴) = 0 ↔ (𝑎‘𝐴) = 0))
155		fveq2 6831	. . . . . 6 ⊢ (𝑝 = 𝑎 → (coeff‘𝑝) = (coeff‘𝑎))
156	155	fveq1d 6833	. . . . 5 ⊢ (𝑝 = 𝑎 → ((coeff‘𝑝)‘(degAA‘𝐴)) = ((coeff‘𝑎)‘(degAA‘𝐴)))
157	156	eqeq1d 2735	. . . 4 ⊢ (𝑝 = 𝑎 → (((coeff‘𝑝)‘(degAA‘𝐴)) = 1 ↔ ((coeff‘𝑎)‘(degAA‘𝐴)) = 1))
158	152, 154, 157	3anbi123d 1438	. . 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 3687	. 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 583	1 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  ∃!wreu 3346  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  {csn 4577   class class class wbr 5095   × cxp 5619   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617  ℂcc 11014  0cc0 11016  1c1 11017   + caddc 11019   · cmul 11021   < clt 11156   − cmin 11354  -cneg 11355   / cdiv 11784  ℕcn 12135  ℕ₀cn0 12391  ℤcz 12478  ℚcq 12856  0_𝑝c0p 25607  Polycply 26126  coeffccoe 26128  degcdgr 26129  𝔸caa 26259  deg_AAcdgraa 43247

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-pm 8762  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-sup 9336  df-inf 9337  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-n0 12392  df-z 12479  df-uz 12743  df-q 12857  df-rp 12901  df-fz 13418  df-fzo 13565  df-fl 13706  df-mod 13784  df-seq 13919  df-exp 13979  df-hash 14248  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-clim 15405  df-rlim 15406  df-sum 15604  df-0p 25608  df-ply 26130  df-coe 26132  df-dgr 26133  df-aa 26260  df-dgraa 43249

This theorem is referenced by:  mpaalem  43259