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Theorem idomrootle 26146

Description: No element of an integral domain can have more than 𝑁 𝑁-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)

**Hypotheses**
Ref	Expression
idomrootle.b	⊢ 𝐵 = (Base‘𝑅)
idomrootle.e	⊢ ↑ = (._g‘(mulGrp‘𝑅))

**Assertion**
Ref	Expression
idomrootle	⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≤ 𝑁)

Distinct variable groups: 𝑦,𝐵 𝑦,𝑁 𝑦,𝑅 𝑦,𝑋 Allowed substitution hint: ↑ (𝑦)

**Proof of Theorem idomrootle**
Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
2		eqid 2737	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
3		eqid 2737	. . 3 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
4		eqid 2737	. . 3 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
5		eqid 2737	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2737	. . 3 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
7		simp1 1137	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ IDomn)
8		isidom 20670	. . . . . . . . 9 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 496	. . . . . . . 8 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
10	7, 9	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ CRing)
11		crngring 20192	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Ring)
13	1	ply1ring 22200	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Poly₁‘𝑅) ∈ Ring)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Ring)
15		ringgrp 20185	. . . . 5 ⊢ ((Poly₁‘𝑅) ∈ Ring → (Poly₁‘𝑅) ∈ Grp)
16	14, 15	syl 17	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Grp)
17		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
18	17	ringmgp 20186	. . . . . . 7 ⊢ ((Poly₁‘𝑅) ∈ Ring → (mulGrp‘(Poly₁‘𝑅)) ∈ Mnd)
19	14, 18	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘(Poly₁‘𝑅)) ∈ Mnd)
20		mndmgm 18678	. . . . . 6 ⊢ ((mulGrp‘(Poly₁‘𝑅)) ∈ Mnd → (mulGrp‘(Poly₁‘𝑅)) ∈ Mgm)
21	19, 20	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘(Poly₁‘𝑅)) ∈ Mgm)
22		simp3 1139	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
23		eqid 2737	. . . . . . 7 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24	23, 1, 2	vr1cl 22170	. . . . . 6 ⊢ (𝑅 ∈ Ring → (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅)))
25	12, 24	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅)))
26	17, 2	mgpbas 20092	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
27		eqid 2737	. . . . . 6 ⊢ (._g‘(mulGrp‘(Poly₁‘𝑅))) = (._g‘(mulGrp‘(Poly₁‘𝑅)))
28	26, 27	mulgnncl 19031	. . . . 5 ⊢ (((mulGrp‘(Poly₁‘𝑅)) ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅))) → (𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)))
29	21, 22, 25, 28	syl3anc 1374	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)))
30		eqid 2737	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
31		idomrootle.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
32	1, 30, 31, 2	ply1sclf 22239	. . . . . 6 ⊢ (𝑅 ∈ Ring → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(Poly₁‘𝑅)))
33	12, 32	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(Poly₁‘𝑅)))
34		simp2 1138	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵)
35	33, 34	ffvelcdmd 7039	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)))
36		eqid 2737	. . . . 5 ⊢ (-_g‘(Poly₁‘𝑅)) = (-_g‘(Poly₁‘𝑅))
37	2, 36	grpsubcl 18962	. . . 4 ⊢ (((Poly₁‘𝑅) ∈ Grp ∧ (𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)) ∧ ((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅))) → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ (Base‘(Poly₁‘𝑅)))
38	16, 29, 35, 37	syl3anc 1374	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ (Base‘(Poly₁‘𝑅)))
39	3, 1, 2	deg1xrcl 26055	. . . . . . . . . 10 ⊢ (((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ ℝ^*)
40	35, 39	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ ℝ^*)
41		0xr 11191	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ^*)
43		nnre 12164	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
44	43	rexrd 11194	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)
45	44	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ^*)
46	3, 1, 31, 30	deg1sclle 26085	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
47	12, 34, 46	syl2anc 585	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
48		nngt0 12188	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
49	48	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
50	40, 42, 45, 47, 49	xrlelttrd 13086	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) < 𝑁)
51	8	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
52		domnnzr 20651	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
53	51, 52	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ NzRing)
54	7, 53	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ NzRing)
55		nnnn0 12420	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
56	55	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
57	3, 1, 23, 17, 27	deg1pw 26094	. . . . . . . . 9 ⊢ ((𝑅 ∈ NzRing ∧ 𝑁 ∈ ℕ₀) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
58	54, 56, 57	syl2anc 585	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
59	50, 58	breqtrrd 5128	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) < ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))))
60	1, 3, 12, 2, 36, 29, 35, 59	deg1sub 26081	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) = ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))))
61	60, 58	eqtrd 2772	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) = 𝑁)
62	61, 56	eqeltrd 2837	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) ∈ ℕ₀)
63	3, 1, 6, 2	deg1nn0clb 26063	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ (Base‘(Poly₁‘𝑅))) → (((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ≠ (0_g‘(Poly₁‘𝑅)) ↔ ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) ∈ ℕ₀))
64	12, 38, 63	syl2anc 585	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ≠ (0_g‘(Poly₁‘𝑅)) ↔ ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) ∈ ℕ₀))
65	62, 64	mpbird 257	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ≠ (0_g‘(Poly₁‘𝑅)))
66	1, 2, 3, 4, 5, 6, 7, 38, 65	fta1g 26143	. 2 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘(^◡((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) “ {(0_g‘𝑅)})) ≤ ((deg₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))))
67		eqid 2737	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
68		eqid 2737	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
69	31	fvexi 6856	. . . . . . . 8 ⊢ 𝐵 ∈ V
70	69	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ V)
71	4, 1, 67, 31	evl1rhm 22288	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
72	10, 71	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
73	2, 68	rhmf 20432	. . . . . . . . 9 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
74	72, 73	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
75	74, 38	ffvelcdmd 7039	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) ∈ (Base‘(𝑅 ↑_s 𝐵)))
76	67, 31, 68, 7, 70, 75	pwselbas 17421	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))):𝐵⟶𝐵)
77	76	ffnd 6671	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) Fn 𝐵)
78		fniniseg2 7016	. . . . 5 ⊢ (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) Fn 𝐵 → (^◡((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) “ {(0_g‘𝑅)}) = {𝑦 ∈ 𝐵 ∣ (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = (0_g‘𝑅)})
79	77, 78	syl 17	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (^◡((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) “ {(0_g‘𝑅)}) = {𝑦 ∈ 𝐵 ∣ (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = (0_g‘𝑅)})
80	10	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
81		simpr 484	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
82	4, 23, 31, 1, 2, 80, 81	evl1vard 22293	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → ((var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅)) ∧ (((eval₁‘𝑅)‘(var₁‘𝑅))‘𝑦) = 𝑦))
83		idomrootle.e	. . . . . . . . . 10 ⊢ ↑ = (._g‘(mulGrp‘𝑅))
84		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑁 ∈ ℕ)
85	84, 55	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑁 ∈ ℕ₀)
86	4, 1, 31, 2, 80, 81, 82, 27, 83, 85	evl1expd 22301	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)) ∧ (((eval₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)))‘𝑦) = (𝑁 ↑ 𝑦)))
87		simpl2 1194	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
88	4, 1, 31, 30, 2, 80, 87, 81	evl1scad 22291	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)) ∧ (((eval₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋))‘𝑦) = 𝑋))
89		eqid 2737	. . . . . . . . 9 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
90	4, 1, 31, 2, 80, 81, 86, 88, 36, 89	evl1subd 22298	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ (Base‘(Poly₁‘𝑅)) ∧ (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = ((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋)))
91	90	simprd 495	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = ((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋))
92	91	eqeq1d 2739	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → ((((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = (0_g‘𝑅) ↔ ((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋) = (0_g‘𝑅)))
93		ringgrp 20185	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
94	12, 93	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Grp)
95	94	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ Grp)
96		eqid 2737	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
97	96	ringmgp 20186	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
98	12, 97	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘𝑅) ∈ Mnd)
99	98	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mnd)
100		mndmgm 18678	. . . . . . . . 9 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm)
101	99, 100	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mgm)
102	96, 31	mgpbas 20092	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
103	102, 83	mulgnncl 19031	. . . . . . . 8 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
104	101, 84, 81, 103	syl3anc 1374	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
105	31, 5, 89	grpsubeq0 18968	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝑁 ↑ 𝑦) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋) = (0_g‘𝑅) ↔ (𝑁 ↑ 𝑦) = 𝑋))
106	95, 104, 87, 105	syl3anc 1374	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋) = (0_g‘𝑅) ↔ (𝑁 ↑ 𝑦) = 𝑋))
107	92, 106	bitrd 279	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → ((((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = (0_g‘𝑅) ↔ (𝑁 ↑ 𝑦) = 𝑋))
108	107	rabbidva 3407	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → {𝑦 ∈ 𝐵 ∣ (((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋)))‘𝑦) = (0_g‘𝑅)} = {𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋})
109	79, 108	eqtrd 2772	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (^◡((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) “ {(0_g‘𝑅)}) = {𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋})
110	109	fveq2d 6846	. 2 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘(^◡((eval₁‘𝑅)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))(-_g‘(Poly₁‘𝑅))((algSc‘(Poly₁‘𝑅))‘𝑋))) “ {(0_g‘𝑅)})) = (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}))
111	66, 110, 61	3brtr3d 5131	1 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≤ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {crab 3401 Vcvv 3442 {csn 4582 class class class wbr 5100 ^◡ccnv 5631 “ cima 5635 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 0cc0 11038 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℕcn 12157 ℕ₀cn0 12413 ♯chash 14265 Basecbs 17148 0_gc0g 17371 ↑_s cpws 17378 Mgmcmgm 18575 Mndcmnd 18671 Grpcgrp 18875 -_gcsg 18877 ._gcmg 19009 mulGrpcmgp 20087 Ringcrg 20180 CRingccrg 20181 RingHom crh 20417 NzRingcnzr 20457 Domncdomn 20637 IDomncidom 20638 algSccascl 21819 var₁cv1 22128 Poly₁cpl1 22129 eval₁ce1 22270 deg₁cdg1 26027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-rhm 20420 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-lmod 20825 df-lss 20895 df-lsp 20935 df-cnfld 21322 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evl1 22272 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107

This theorem is referenced by: unitscyglem5 42563 idomodle 43542

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