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Theorem idomrootle 26212

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 20725	. . . . . . . . 9 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 497	. . . . . . . 8 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
10	7, 9	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ CRing)
11		crngring 20242	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Ring)
13	1	ply1ring 22249	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Poly₁‘𝑅) ∈ Ring)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Ring)
15		ringgrp 20235	. . . . 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 20236	. . . . . . 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 18754	. . . . . 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 22219	. . . . . 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 20142	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
27		eqid 2737	. . . . . 6 ⊢ (._g‘(mulGrp‘(Poly₁‘𝑅))) = (._g‘(mulGrp‘(Poly₁‘𝑅)))
28	26, 27	mulgnncl 19107	. . . . 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 1373	. . . 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 22288	. . . . . 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 7105	. . . 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 19038	. . . 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 1373	. . 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 26121	. . . . . . . . . 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 11308	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ^*)
43		nnre 12273	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
44	43	rexrd 11311	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)
45	44	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ^*)
46	3, 1, 31, 30	deg1sclle 26151	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
47	12, 34, 46	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
48		nngt0 12297	. . . . . . . . . 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 13202	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) < 𝑁)
51	8	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
52		domnnzr 20706	. . . . . . . . . . 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 12533	. . . . . . . . . 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 26160	. . . . . . . . 9 ⊢ ((𝑅 ∈ NzRing ∧ 𝑁 ∈ ℕ₀) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
58	54, 56, 57	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
59	50, 58	breqtrrd 5171	. . . . . . 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 26147	. . . . . 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 2777	. . . . 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 2841	. . . 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 26129	. . . . 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 584	. . . 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 26209	. 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 6920	. . . . . . . 8 ⊢ 𝐵 ∈ V
70	69	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ V)
71	4, 1, 67, 31	evl1rhm 22336	. . . . . . . . . 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 20485	. . . . . . . . 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 7105	. . . . . . 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 17534	. . . . . 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 6737	. . . . 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 7082	. . . . 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 22341	. . . . . . . . . 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 1194	. . . . . . . . . . 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 22349	. . . . . . . . 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 1193	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
88	4, 1, 31, 30, 2, 80, 87, 81	evl1scad 22339	. . . . . . . . 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 22346	. . . . . . . 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 20235	. . . . . . . . 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 20236	. . . . . . . . . . 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 18754	. . . . . . . . 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 20142	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
103	102, 83	mulgnncl 19107	. . . . . . . 8 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
104	101, 84, 81, 103	syl3anc 1373	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
105	31, 5, 89	grpsubeq0 19044	. . . . . . 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 1373	. . . . . 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 3443	. . . 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 2777	. . 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 6910	. 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 5174	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 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 Vcvv 3480 {csn 4626 class class class wbr 5143 ^◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℕ₀cn0 12526 ♯chash 14369 Basecbs 17247 0_gc0g 17484 ↑_s cpws 17491 Mgmcmgm 18651 Mndcmnd 18747 Grpcgrp 18951 -_gcsg 18953 ._gcmg 19085 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 NzRingcnzr 20512 Domncdomn 20692 IDomncidom 20693 algSccascl 21872 var₁cv1 22177 Poly₁cpl1 22178 eval₁ce1 22318 deg₁cdg1 26093

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-lmod 20860 df-lss 20930 df-lsp 20970 df-cnfld 21365 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-evl1 22320 df-mdeg 26094 df-deg1 26095 df-mon1 26170 df-uc1p 26171 df-q1p 26172 df-r1p 26173

This theorem is referenced by: unitscyglem5 42200 idomodle 43203

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