Theorem irngnminplynz 34400

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 21063	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
4		eqid 2763	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
5	4	subrgring 20841	. . . . 5 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ Ring)
6	3, 5	syl 18	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
7		eqid 2763	. . . . 5 ⊢ (Poly1‘(𝐸 ↾s 𝐹)) = (Poly1‘(𝐸 ↾s 𝐹))
8	7	ply1ring 22580	. . . 4 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
9	6, 8	syl 18	. . 3 ⊢ (𝜑 → (Poly1‘(𝐸 ↾s 𝐹)) ∈ Ring)
10		eqid 2763	. . . . . 6 ⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
11		eqid 2763	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
12		irngnminplynz.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
13	12	fldcrngd 21010	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing)
14		eqid 2763	. . . . . . . 8 ⊢ (0g‘𝐸) = (0g‘𝐸)
15	10, 4, 11, 14, 13, 3	irngssv 34376	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
16		irngnminplynz.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
17	15, 16	sseldd 3914	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
18		eqid 2763	. . . . . 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 34390	. . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))))
20		eqid 2763	. . . . . 6 ⊢ (Base‘(Poly1‘(𝐸 ↾s 𝐹))) = (Base‘(Poly1‘(𝐸 ↾s 𝐹)))
21		eqid 2763	. . . . . 6 ⊢ (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹))) = (LIdeal‘(Poly1‘(𝐸 ↾s 𝐹)))
22	20, 21	lidlss 21505	. . . . 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 18	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ⊆ (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
24	4	sdrgdrng 21062	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
25	1, 24	syl 18	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
26		eqid 2763	. . . . . 6 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
27	7, 26, 21	ig1pcl 26512	. . . . 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 596	. . . 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 3914	. . 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 2763	. . . . 5 ⊢ (RSpan‘(Poly1‘(𝐸 ↾s 𝐹))) = (RSpan‘(Poly1‘(𝐸 ↾s 𝐹)))
31	10, 7, 11, 12, 1, 17, 14, 18, 30, 26	ply1annig1p 34392	. . . 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 6896	. . . . . . . . 9 ⊢ (𝑞 = 𝑝 → ((𝐸 evalSub1 𝐹)‘𝑞) = ((𝐸 evalSub1 𝐹)‘𝑝))
33	32	fveq1d 6898	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴))
34	33	eqeq1d 2765	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸) ↔ (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸)))
35		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍}))
36	35	eldifad 4054	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ∈ dom (𝐸 evalSub1 𝐹))
37	13	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝐸 ∈ CRing)
38	3	ad2antrr 739	. . . . . . . . . . 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 34149	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐸 evalSub1 𝐹) = (Base‘(Poly1‘(𝐸 ↾s 𝐹))))
40	39	ad2antrr 739	. . . . . . . . . . . 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 2866	. . . . . . . . . . 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 34150	. . . . . . . . . 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 6721	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ((𝐸 evalSub1 𝐹)‘𝑝) Fn (Base‘𝐸))
44		elpreima 7068	. . . . . . . . . 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 505	. . . . . . . . 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 600	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)})
47		elsni 4601	. . . . . . . 8 ⊢ ((((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) ∈ {(0g‘𝐸)} → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝐴) = (0g‘𝐸))
48	46, 47	syl 18	. . . . . . 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 3650	. . . . . 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 4753	. . . . . . . . 9 ⊢ (𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍}) → 𝑝 ≠ 𝑍)
51	35, 50	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ 𝑍)
52		eqid 2763	. . . . . . . . . 10 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
53		irngnminplynz.z	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘(Poly1‘𝐸))
54	52, 4, 7, 20, 3, 53	ressply10g 34155	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
55	54	ad2antrr 739	. . . . . . . 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 3028	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → 𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))))
57		nelsn 4627	. . . . . . 7 ⊢ (𝑝 ≠ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) → ¬ 𝑝 ∈ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
58	56, 57	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})) ∧ 𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)})) → ¬ 𝑝 ∈ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
59		nelne1 3056	. . . . . 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 596	. . . . 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 34379	. . . . . . 7 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
62	16, 61	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})(◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
63		eliun 4958	. . . . . 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 221	. . . . 5 ⊢ (𝜑 → ∃𝑝 ∈ (dom (𝐸 evalSub1 𝐹) ∖ {𝑍})𝐴 ∈ (◡((𝐸 evalSub1 𝐹)‘𝑝) “ {(0g‘𝐸)}))
65	60, 64	r19.29a 3174	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ {(0g‘(Poly1‘(𝐸 ↾s 𝐹)))})
66	31, 65	eqnetrrd 3027	. . 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 2763	. . . . 5 ⊢ (0g‘(Poly1‘(𝐸 ↾s 𝐹))) = (0g‘(Poly1‘(𝐸 ↾s 𝐹)))
68	20, 67, 30	pidlnzb 34028	. . . 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 483	. . 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 851	. 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 34393	. 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 3036	1 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  ∪ ciun 4954  ^◡ccnv 5654  dom cdm 5655   “ cima 5658   Fn wfn 6542  ‘cfv 6547  (class class class)co 7430  Basecbs 17402   ↾_s cress 17423  0_gc0g 17625  Ringcrg 20474  CRingccrg 20475  SubRingcsubrg 20836  DivRingcdr 20995  Fieldcfield 20996  SubDRingcsdrg 21058  LIdealclidl 21499  RSpancrsp 21500  Poly₁cpl1 22510   evalSub₁ ces1 22646  idlGen_1pcig1p 26463   IntgRing cirng 34371   minPoly cminply 34387

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291  ax-addf 11292

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7703  df-ofr 7704  df-om 7888  df-1st 8011  df-2nd 8012  df-supp 8191  df-tpos 8257  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-er 8731  df-map 8863  df-pm 8864  df-ixp 8940  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-fsupp 9368  df-sup 9447  df-inf 9448  df-oi 9517  df-card 10033  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-fzo 13802  df-seq 14158  df-hash 14488  df-struct 17340  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-ress 17424  df-plusg 17456  df-mulr 17457  df-starv 17458  df-sca 17459  df-vsca 17460  df-ip 17461  df-tset 17462  df-ple 17463  df-ds 17465  df-unif 17466  df-hom 17467  df-cco 17468  df-0g 17627  df-gsum 17628  df-prds 17633  df-pws 17635  df-mre 17771  df-mrc 17772  df-acs 17774  df-mgm 18831  df-sgrp 18923  df-mnd 18939  df-mhm 18993  df-submnd 18994  df-grp 19162  df-minusg 19163  df-sbg 19164  df-mulg 19293  df-subg 19348  df-ghm 19443  df-cntz 19546  df-cmn 20011  df-abl 20012  df-mgp 20376  df-rng 20390  df-ur 20423  df-srg 20428  df-ring 20476  df-cring 20477  df-oppr 20582  df-dvdsr 20602  df-unit 20603  df-invr 20633  df-rhm 20717  df-subrng 20813  df-subrg 20837  df-rlreg 20961  df-drng 20997  df-field 20998  df-sdrg 21059  df-lmod 21152  df-lss 21222  df-lsp 21262  df-sra 21463  df-rgmod 21464  df-lidl 21501  df-rsp 21502  df-cnfld 21694  df-assa 22176  df-asp 22177  df-ascl 22178  df-psr 22232  df-mvr 22233  df-mpl 22234  df-opsr 22236  df-evls 22398  df-evl 22399  df-psr1 22513  df-vr1 22514  df-ply1 22515  df-coe1 22516  df-evls1 22648  df-evl1 22649  df-mdeg 26388  df-deg1 26389  df-mon1 26464  df-uc1p 26465  df-q1p 26466  df-r1p 26467  df-ig1p 26468  df-irng 34372  df-minply 34388

This theorem is used by:  minplym1p  34401  irredminply  34404  algextdeglem4  34408  algextdeglem7  34411  algextdeglem8  34412