Theorem irngnminplynz 33705

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 20814	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
4		eqid 2740	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5	4	subrgring 20602	. . . . 5 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
7		eqid 2740	. . . . 5 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
8	7	ply1ring 22270	. . . 4 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
10		eqid 2740	. . . . . 6 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
11		eqid 2740	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
12		irngnminplynz.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
13	12	fldcrngd 20764	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing)
14		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝐸) = (0g‘𝐸)
15	10, 4, 11, 14, 13, 3	irngssv 33688	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		irngnminplynz.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 4009	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2740	. . . . . 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 33695	. . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))))
20		eqid 2740	. . . . . 6 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
21		eqid 2740	. . . . . 6 ⊢ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) = (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))
22	20, 21	lidlss 21245	. . . . 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 20813	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
26		eqid 2740	. . . . . 6 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
27	7, 26, 21	ig1pcl 26238	. . . . 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 4009	. . 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 2740	. . . . 5 ⊢ (RSpan‘(Poly1‘(𝐸 ↾s 𝐹))) = (RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))
31	10, 7, 11, 12, 1, 17, 14, 18, 30, 26	ply1annig1p 33697	. . . 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 6920	. . . . . . . . 9 ⊢ (𝑞 = 𝑝 → ((𝐸 evalSub1 𝐹)‘𝑞) = ((𝐸 evalSub1 𝐹)‘𝑝))
33	32	fveq1d 6922	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴))
34	33	eqeq1d 2742	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸) ↔ (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸)))
35		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍}))
36	35	eldifad 3988	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ dom (𝐸 evalSub1 𝐹))
37	13	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝐸 ∈ CRing)
38	3	ad2antrr 725	. . . . . . . . . . 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 33552	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐸 evalSub1 𝐹) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
40	39	ad2antrr 725	. . . . . . . . . . . 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 2846	. . . . . . . . . . 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 33553	. . . . . . . . . 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 6748	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸))
44		elpreima 7091	. . . . . . . . . 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 4665	. . . . . . . 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 3710	. . . . . 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 4815	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
53		irngnminplynz.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
54	52, 4, 7, 20, 3, 53	ressply10g 33557	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
55	54	ad2antrr 725	. . . . . . . 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 3016	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
57		nelsn 4688	. . . . . . 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 3045	. . . . . 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 33691	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
62	16, 61	eleqtrd 2846	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
63		eliun 5019	. . . . . 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 218	. . . . 5 ⊢ (𝜑 → ∃𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
65	60, 64	r19.29a 3168	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
66	31, 65	eqnetrrd 3015	. . 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 2740	. . . . 5 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
68	20, 67, 30	pidlnzb 33415	. . . 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 837	. 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 33698	. 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 3024	1 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ∪ ciun 5015  ^◡ccnv 5699  dom cdm 5700   “ cima 5703   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  0_gc0g 17499  Ringcrg 20260  CRingccrg 20261  SubRingcsubrg 20595  DivRingcdr 20751  Fieldcfield 20752  SubDRingcsdrg 20809  LIdealclidl 21239  RSpancrsp 21240  Poly₁cpl1 22199   evalSub₁ ces1 22338  idlGen_1pcig1p 26189   IntgRing cirng 33683   minPoly cminply 33692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-rhm 20498  df-subrng 20572  df-subrg 20597  df-rlreg 20716  df-drng 20753  df-field 20754  df-sdrg 20810  df-lmod 20882  df-lss 20953  df-lsp 20993  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-rsp 21242  df-cnfld 21388  df-assa 21896  df-asp 21897  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-evls 22121  df-evl 22122  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-evls1 22340  df-evl1 22341  df-mdeg 26114  df-deg1 26115  df-mon1 26190  df-uc1p 26191  df-q1p 26192  df-r1p 26193  df-ig1p 26194  df-irng 33684  df-minply 33693

This theorem is referenced by:  minplym1p  33706  irredminply  33707  algextdeglem4  33711  algextdeglem7  33714  algextdeglem8  33715