Theorem irngnminplynz 33702

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 20700	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
4		eqid 2729	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5	4	subrgring 20483	. . . . 5 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
7		eqid 2729	. . . . 5 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
8	7	ply1ring 22132	. . . 4 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
10		eqid 2729	. . . . . 6 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
11		eqid 2729	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
12		irngnminplynz.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
13	12	fldcrngd 20651	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing)
14		eqid 2729	. . . . . . . 8 ⊢ (0g‘𝐸) = (0g‘𝐸)
15	10, 4, 11, 14, 13, 3	irngssv 33683	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		irngnminplynz.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 3947	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2729	. . . . . 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 33692	. . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))))
20		eqid 2729	. . . . . 6 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
21		eqid 2729	. . . . . 6 ⊢ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) = (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))
22	20, 21	lidlss 21122	. . . . 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 20699	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
26		eqid 2729	. . . . . 6 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
27	7, 26, 21	ig1pcl 26084	. . . . 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 584	. . . 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 3947	. . 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 2729	. . . . 5 ⊢ (RSpan‘(Poly1‘(𝐸 ↾s 𝐹))) = (RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))
31	10, 7, 11, 12, 1, 17, 14, 18, 30, 26	ply1annig1p 33694	. . . 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 6858	. . . . . . . . 9 ⊢ (𝑞 = 𝑝 → ((𝐸 evalSub1 𝐹)‘𝑞) = ((𝐸 evalSub1 𝐹)‘𝑝))
33	32	fveq1d 6860	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴))
34	33	eqeq1d 2731	. . . . . . 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 3926	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ dom (𝐸 evalSub1 𝐹))
37	13	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝐸 ∈ CRing)
38	3	ad2antrr 726	. . . . . . . . . . 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 33530	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐸 evalSub1 𝐹) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
40	39	ad2antrr 726	. . . . . . . . . . . 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 2830	. . . . . . . . . . 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 33531	. . . . . . . . . 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 6689	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸))
44		elpreima 7030	. . . . . . . . . 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 588	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)})
47		elsni 4606	. . . . . . . 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 3661	. . . . . 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 4754	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
53		irngnminplynz.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
54	52, 4, 7, 20, 3, 53	ressply10g 33536	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
55	54	ad2antrr 726	. . . . . . . 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 2994	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
57		nelsn 4630	. . . . . . 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 3022	. . . . . 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 584	. . . . 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 33686	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
62	16, 61	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
63		eliun 4959	. . . . . 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 3141	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
66	31, 65	eqnetrrd 2993	. . 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 2729	. . . . 5 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
68	20, 67, 30	pidlnzb 33393	. . . 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 33695	. 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 3002	1 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3405   ∖ cdif 3911   ⊆ wss 3914  {csn 4589  ∪ ciun 4955  ^◡ccnv 5637  dom cdm 5638   “ cima 5641   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402  Ringcrg 20142  CRingccrg 20143  SubRingcsubrg 20478  DivRingcdr 20638  Fieldcfield 20639  SubDRingcsdrg 20695  LIdealclidl 21116  RSpancrsp 21117  Poly₁cpl1 22061   evalSub₁ ces1 22200  idlGen_1pcig1p 26035   IntgRing cirng 33678   minPoly cminply 33689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-rhm 20381  df-subrng 20455  df-subrg 20479  df-rlreg 20603  df-drng 20640  df-field 20641  df-sdrg 20696  df-lmod 20768  df-lss 20838  df-lsp 20878  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-cnfld 21265  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203  df-mdeg 25960  df-deg1 25961  df-mon1 26036  df-uc1p 26037  df-q1p 26038  df-r1p 26039  df-ig1p 26040  df-irng 33679  df-minply 33690

This theorem is referenced by:  minplym1p  33703  irredminply  33706  algextdeglem4  33710  algextdeglem7  33713  algextdeglem8  33714