Theorem irngnminplynz 33709

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 20707	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
4		eqid 2730	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5	4	subrgring 20490	. . . . 5 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
7		eqid 2730	. . . . 5 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
8	7	ply1ring 22139	. . . 4 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
10		eqid 2730	. . . . . 6 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
11		eqid 2730	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
12		irngnminplynz.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
13	12	fldcrngd 20658	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing)
14		eqid 2730	. . . . . . . 8 ⊢ (0g‘𝐸) = (0g‘𝐸)
15	10, 4, 11, 14, 13, 3	irngssv 33690	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		irngnminplynz.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 3950	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2730	. . . . . 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 33699	. . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))))
20		eqid 2730	. . . . . 6 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
21		eqid 2730	. . . . . 6 ⊢ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) = (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))
22	20, 21	lidlss 21129	. . . . 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 20706	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
26		eqid 2730	. . . . . 6 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
27	7, 26, 21	ig1pcl 26091	. . . . 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 3950	. . 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 2730	. . . . 5 ⊢ (RSpan‘(Poly1‘(𝐸 ↾s 𝐹))) = (RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))
31	10, 7, 11, 12, 1, 17, 14, 18, 30, 26	ply1annig1p 33701	. . . 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 6861	. . . . . . . . 9 ⊢ (𝑞 = 𝑝 → ((𝐸 evalSub1 𝐹)‘𝑞) = ((𝐸 evalSub1 𝐹)‘𝑝))
33	32	fveq1d 6863	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴))
34	33	eqeq1d 2732	. . . . . . 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 3929	. . . . . . 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 33537	. . . . . . . . . . . . 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 2831	. . . . . . . . . . 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 33538	. . . . . . . . . 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 6692	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸))
44		elpreima 7033	. . . . . . . . . 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 4609	. . . . . . . 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 3664	. . . . . 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 4757	. . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
53		irngnminplynz.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
54	52, 4, 7, 20, 3, 53	ressply10g 33543	. . . . . . . . 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 2995	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
57		nelsn 4633	. . . . . . 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 3023	. . . . . 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 33693	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
62	16, 61	eleqtrd 2831	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
63		eliun 4962	. . . . . 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 3142	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
66	31, 65	eqnetrrd 2994	. . 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 2730	. . . . 5 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
68	20, 67, 30	pidlnzb 33400	. . . 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 33702	. 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 3003	1 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  {crab 3408   ∖ cdif 3914   ⊆ wss 3917  {csn 4592  ∪ ciun 4958  ^◡ccnv 5640  dom cdm 5641   “ cima 5644   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  0_gc0g 17409  Ringcrg 20149  CRingccrg 20150  SubRingcsubrg 20485  DivRingcdr 20645  Fieldcfield 20646  SubDRingcsdrg 20702  LIdealclidl 21123  RSpancrsp 21124  Poly₁cpl1 22068   evalSub₁ ces1 22207  idlGen_1pcig1p 26042   IntgRing cirng 33685   minPoly cminply 33696

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-rhm 20388  df-subrng 20462  df-subrg 20486  df-rlreg 20610  df-drng 20647  df-field 20648  df-sdrg 20703  df-lmod 20775  df-lss 20845  df-lsp 20885  df-sra 21087  df-rgmod 21088  df-lidl 21125  df-rsp 21126  df-cnfld 21272  df-assa 21769  df-asp 21770  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-evls 21988  df-evl 21989  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-evls1 22209  df-evl1 22210  df-mdeg 25967  df-deg1 25968  df-mon1 26043  df-uc1p 26044  df-q1p 26045  df-r1p 26046  df-ig1p 26047  df-irng 33686  df-minply 33697

This theorem is referenced by:  minplym1p  33710  irredminply  33713  algextdeglem4  33717  algextdeglem7  33720  algextdeglem8  33721