Theorem irngnminplynz 33075

Description: Integral elements have nonzero minimal polynomials. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypotheses

Ref	Expression
irngnminplynz.z	⊢ 𝑍 = (0g‘(Poly1‘𝐸))
irngnminplynz.e	⊢ (𝜑 → 𝐸 ∈ Field)
irngnminplynz.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
irngnminplynz.m	⊢ 𝑀 = (𝐸 minPoly 𝐹)
irngnminplynz.a	⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))

Assertion

Ref	Expression
irngnminplynz	⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Proof of Theorem irngnminplynz

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irngnminplynz.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
2		sdrgsubrg 20554	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
4		eqid 2731	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5	4	subrgring 20468	. . . . 5 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
7		eqid 2731	. . . . 5 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
8	7	ply1ring 22003	. . . 4 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
10		eqid 2731	. . . . . 6 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
11		eqid 2731	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
12		irngnminplynz.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
13	12	fldcrngd 20517	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing)
14		eqid 2731	. . . . . . . 8 ⊢ (0g‘𝐸) = (0g‘𝐸)
15	10, 4, 11, 14, 13, 3	irngssv 33056	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		irngnminplynz.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 3983	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2731	. . . . . 6 ⊢ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}
19	10, 7, 11, 13, 3, 17, 14, 18	ply1annidl 33067	. . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))))
20		eqid 2731	. . . . . 6 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
21		eqid 2731	. . . . . 6 ⊢ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) = (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))
22	20, 21	lidlss 20982	. . . . 5 ⊢ ({𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ⊆ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
23	19, 22	syl 17	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ⊆ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
24	4	sdrgdrng 20553	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
26		eqid 2731	. . . . . 6 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
27	7, 26, 21	ig1pcl 25942	. . . . 5 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))) → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})
28	25, 19, 27	syl2anc 583	. . . 4 ⊢ (𝜑 → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})
29	23, 28	sseldd 3983	. . 3 ⊢ (𝜑 → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
30		eqid 2731	. . . . 5 ⊢ (RSpan‘(Poly1‘(𝐸 ↾s 𝐹))) = (RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))
31	10, 7, 11, 12, 1, 17, 14, 18, 30, 26	ply1annig1p 33069	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} = ((RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})}))
32		fveq2 6891	. . . . . . . . 9 ⊢ (𝑞 = 𝑝 → ((𝐸 evalSub1 𝐹)‘𝑞) = ((𝐸 evalSub1 𝐹)‘𝑝))
33	32	fveq1d 6893	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴))
34	33	eqeq1d 2733	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸) ↔ (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸)))
35		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍}))
36	35	eldifad 3960	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ dom (𝐸 evalSub1 𝐹))
37	13	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝐸 ∈ CRing)
38	3	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝐹 ∈ (SubRing‘𝐸))
39	10, 7, 20, 13, 3	evls1dm 32930	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐸 evalSub1 𝐹) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
40	39	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → dom (𝐸 evalSub1 𝐹) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
41	36, 40	eleqtrd 2834	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
42	10, 7, 20, 37, 38, 11, 41	evls1fvf 32931	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝):(Base‘𝐸)⟶(Base‘𝐸))
43	42	ffnd 6718	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸))
44		elpreima 7059	. . . . . . . . . 10 ⊢ (((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸) → (𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}) ↔ (𝐴 ∈ (Base‘𝐸) ∧ (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)})))
45	44	simplbda 499	. . . . . . . . 9 ⊢ ((((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)})
46	43, 45	sylancom 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)})
47		elsni 4645	. . . . . . . 8 ⊢ ((((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)} → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸))
48	46, 47	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸))
49	34, 36, 48	elrabd 3685	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})
50		eldifsni 4793	. . . . . . . . 9 ⊢ (𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍}) → 𝑝 ≠ 𝑍)
51	35, 50	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ 𝑍)
52		eqid 2731	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
53		irngnminplynz.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
54	52, 4, 7, 20, 3, 53	ressply10g 32945	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
55	54	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
56	51, 55	neeqtrd 3009	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
57		nelsn 4668	. . . . . . 7 ⊢ (𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) → ¬ 𝑝 ∈ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ¬ 𝑝 ∈ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
59		nelne1 3038	. . . . . 6 ⊢ ((𝑝 ∈ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∧ ¬ 𝑝 ∈ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))}) → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
60	49, 58, 59	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
61	10, 53, 14, 12, 1	irngnzply1 33059	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
62	16, 61	eleqtrd 2834	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
63		eliun 5001	. . . . . 6 ⊢ (𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}) ↔ ∃𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
64	62, 63	sylib 217	. . . . 5 ⊢ (𝜑 → ∃𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
65	60, 64	r19.29a 3161	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
66	31, 65	eqnetrrd 3008	. . 3 ⊢ (𝜑 → ((RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})}) ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
67		eqid 2731	. . . . 5 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
68	20, 67, 30	pidlnzb 32829	. . . 4 ⊢ (((Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring ∧ ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹)))) → (((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) ↔ ((RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})}) ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))}))
69	68	biimpar 477	. . 3 ⊢ ((((Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring ∧ ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Base‘(Poly1‘(𝐸 ↾s 𝐹)))) ∧ ((RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)})}) ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))}) → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
70	9, 29, 66, 69	syl21anc 835	. 2 ⊢ (𝜑 → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}) ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
71		irngnminplynz.m	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
72	10, 7, 11, 12, 1, 17, 14, 18, 30, 26, 71	minplyval 33070	. 2 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)}))
73	70, 72, 54	3netr4d 3017	1 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  {crab 3431   ∖ cdif 3945   ⊆ wss 3948  {csn 4628  ∪ ciun 4997  ^◡ccnv 5675  dom cdm 5676   “ cima 5679   Fn wfn 6538  ‘cfv 6543  (class class class)co 7412  Basecbs 17151   ↾_s cress 17180  0_gc0g 17392  Ringcrg 20131  CRingccrg 20132  SubRingcsubrg 20461  DivRingcdr 20504  Fieldcfield 20505  SubDRingcsdrg 20549  LIdealclidl 20932  RSpancrsp 20933  Poly₁cpl1 21933   evalSub₁ ces1 22065  idlGen_1pcig1p 25896   IntgRing cirng 33051   minPoly cminply 33060

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195  ax-mulf 11196

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18568  df-sgrp 18647  df-mnd 18663  df-mhm 18708  df-submnd 18709  df-grp 18861  df-minusg 18862  df-sbg 18863  df-mulg 18991  df-subg 19043  df-ghm 19132  df-cntz 19226  df-cmn 19695  df-abl 19696  df-mgp 20033  df-rng 20051  df-ur 20080  df-srg 20085  df-ring 20133  df-cring 20134  df-oppr 20229  df-dvdsr 20252  df-unit 20253  df-invr 20283  df-rhm 20367  df-subrng 20438  df-subrg 20463  df-drng 20506  df-field 20507  df-sdrg 20550  df-lmod 20620  df-lss 20691  df-lsp 20731  df-sra 20934  df-rgmod 20935  df-lidl 20936  df-rsp 20937  df-rlreg 21103  df-cnfld 21149  df-assa 21631  df-asp 21632  df-ascl 21633  df-psr 21685  df-mvr 21686  df-mpl 21687  df-opsr 21689  df-evls 21859  df-evl 21860  df-psr1 21936  df-vr1 21937  df-ply1 21938  df-coe1 21939  df-evls1 22067  df-evl1 22068  df-mdeg 25819  df-deg1 25820  df-mon1 25897  df-uc1p 25898  df-q1p 25899  df-r1p 25900  df-ig1p 25901  df-irng 33052  df-minply 33061

This theorem is referenced by:  minplym1p  33076  algextdeglem4  33080  algextdeglem7  33083  algextdeglem8  33084