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Theorem idomrootle 26078

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 2729	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
2		eqid 2729	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
3		eqid 2729	. . 3 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
4		eqid 2729	. . 3 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
5		eqid 2729	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2729	. . 3 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
7		simp1 1136	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ IDomn)
8		isidom 20634	. . . . . . . . 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 20154	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Ring)
13	1	ply1ring 22132	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Poly₁‘𝑅) ∈ Ring)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Ring)
15		ringgrp 20147	. . . . 5 ⊢ ((Poly₁‘𝑅) ∈ Ring → (Poly₁‘𝑅) ∈ Grp)
16	14, 15	syl 17	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Grp)
17		eqid 2729	. . . . . . . 8 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
18	17	ringmgp 20148	. . . . . . 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 18668	. . . . . 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 1138	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
23		eqid 2729	. . . . . . 7 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24	23, 1, 2	vr1cl 22102	. . . . . 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 20054	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
27		eqid 2729	. . . . . 6 ⊢ (._g‘(mulGrp‘(Poly₁‘𝑅))) = (._g‘(mulGrp‘(Poly₁‘𝑅)))
28	26, 27	mulgnncl 19021	. . . . 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 2729	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
31		idomrootle.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
32	1, 30, 31, 2	ply1sclf 22171	. . . . . 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 1137	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵)
35	33, 34	ffvelcdmd 7057	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)))
36		eqid 2729	. . . . 5 ⊢ (-_g‘(Poly₁‘𝑅)) = (-_g‘(Poly₁‘𝑅))
37	2, 36	grpsubcl 18952	. . . 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 25987	. . . . . . . . . 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 11221	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ^*)
43		nnre 12193	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
44	43	rexrd 11224	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)
45	44	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ^*)
46	3, 1, 31, 30	deg1sclle 26017	. . . . . . . . . 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 12217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
49	48	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
50	40, 42, 45, 47, 49	xrlelttrd 13120	. . . . . . . 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 20615	. . . . . . . . . . 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 12449	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
56	55	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
57	3, 1, 23, 17, 27	deg1pw 26026	. . . . . . . . 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 5135	. . . . . . 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 26013	. . . . . 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 2764	. . . . 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 2828	. . . 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 25995	. . . . 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 26075	. 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 2729	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
68		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
69	31	fvexi 6872	. . . . . . . 8 ⊢ 𝐵 ∈ V
70	69	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ V)
71	4, 1, 67, 31	evl1rhm 22219	. . . . . . . . . 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 20394	. . . . . . . . 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 7057	. . . . . . 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 17452	. . . . . 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 6689	. . . . 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 7034	. . . . 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 22224	. . . . . . . . . 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 22232	. . . . . . . . 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 22222	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
90	4, 1, 31, 2, 80, 81, 86, 88, 36, 89	evl1subd 22229	. . . . . . . 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 2731	. . . . . 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 20147	. . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
97	96	ringmgp 20148	. . . . . . . . . . 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 18668	. . . . . . . . 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 20054	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
103	102, 83	mulgnncl 19021	. . . . . . . 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 18958	. . . . . . 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 3412	. . . 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 2764	. . 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 6862	. 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 5138	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3405 Vcvv 3447 {csn 4589 class class class wbr 5107 ^◡ccnv 5637 “ cima 5641 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 0cc0 11068 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℕcn 12186 ℕ₀cn0 12442 ♯chash 14295 Basecbs 17179 0_gc0g 17402 ↑_s cpws 17409 Mgmcmgm 18565 Mndcmnd 18661 Grpcgrp 18865 -_gcsg 18867 ._gcmg 18999 mulGrpcmgp 20049 Ringcrg 20142 CRingccrg 20143 RingHom crh 20378 NzRingcnzr 20421 Domncdomn 20601 IDomncidom 20602 algSccascl 21761 var₁cv1 22060 Poly₁cpl1 22061 eval₁ce1 22201 deg₁cdg1 25959

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-oadd 8438 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-oi 9463 df-dju 9854 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-xnn0 12516 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-nzr 20422 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-domn 20604 df-idom 20605 df-lmod 20768 df-lss 20838 df-lsp 20878 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-evl1 22203 df-mdeg 25960 df-deg1 25961 df-mon1 26036 df-uc1p 26037 df-q1p 26038 df-r1p 26039

This theorem is referenced by: unitscyglem5 42187 idomodle 43180

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