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Theorem idomrootle 26123

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 2725	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
2		eqid 2725	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
3		eqid 2725	. . 3 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘𝑅)
4		eqid 2725	. . 3 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
5		eqid 2725	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2725	. . 3 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
7		simp1 1133	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ IDomn)
8		isidom 21256	. . . . . . . . 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 20187	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Ring)
13	1	ply1ring 22173	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Poly₁‘𝑅) ∈ Ring)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Ring)
15		ringgrp 20180	. . . . 5 ⊢ ((Poly₁‘𝑅) ∈ Ring → (Poly₁‘𝑅) ∈ Grp)
16	14, 15	syl 17	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Grp)
17		eqid 2725	. . . . . . . 8 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
18	17	ringmgp 20181	. . . . . . 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 18698	. . . . . 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 1135	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
23		eqid 2725	. . . . . . 7 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24	23, 1, 2	vr1cl 22143	. . . . . 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 20082	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
27		eqid 2725	. . . . . 6 ⊢ (._g‘(mulGrp‘(Poly₁‘𝑅))) = (._g‘(mulGrp‘(Poly₁‘𝑅)))
28	26, 27	mulgnncl 19046	. . . . 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 1368	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)))
30		eqid 2725	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
31		idomrootle.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
32	1, 30, 31, 2	ply1sclf 22211	. . . . . 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 1134	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵)
35	33, 34	ffvelcdmd 7089	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)))
36		eqid 2725	. . . . 5 ⊢ (-_g‘(Poly₁‘𝑅)) = (-_g‘(Poly₁‘𝑅))
37	2, 36	grpsubcl 18978	. . . 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 1368	. . 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 26034	. . . . . . . . . 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 11289	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ^*)
43		nnre 12247	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
44	43	rexrd 11292	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)
45	44	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ^*)
46	3, 1, 31, 30	deg1sclle 26064	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( deg₁ ‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
47	12, 34, 46	syl2anc 582	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (( deg₁ ‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
48		nngt0 12271	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
49	48	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
50	40, 42, 45, 47, 49	xrlelttrd 13169	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (( deg₁ ‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) < 𝑁)
51	8	simprbi 495	. . . . . . . . . . 11 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
52		domnnzr 21244	. . . . . . . . . . 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 12507	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
56	55	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
57	3, 1, 23, 17, 27	deg1pw 26072	. . . . . . . . 9 ⊢ ((𝑅 ∈ NzRing ∧ 𝑁 ∈ ℕ₀) → (( deg₁ ‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
58	54, 56, 57	syl2anc 582	. . . . . . . 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 26060	. . . . . 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 2765	. . . . 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 2825	. . . 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 26042	. . . . 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 582	. . . 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 256	. . 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 26120	. 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 2725	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
68		eqid 2725	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
69	31	fvexi 6905	. . . . . . . 8 ⊢ 𝐵 ∈ V
70	69	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ V)
71	4, 1, 67, 31	evl1rhm 22258	. . . . . . . . . 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 20426	. . . . . . . . 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 7089	. . . . . . 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 17468	. . . . . 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 6717	. . . . 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 7065	. . . . 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 479	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
81		simpr 483	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
82	4, 23, 31, 1, 2, 80, 81	evl1vard 22263	. . . . . . . . . 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 1190	. . . . . . . . . . 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 22271	. . . . . . . . 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 1189	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
88	4, 1, 31, 30, 2, 80, 87, 81	evl1scad 22261	. . . . . . . . 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 2725	. . . . . . . . 9 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
90	4, 1, 31, 2, 80, 81, 86, 88, 36, 89	evl1subd 22268	. . . . . . . 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 494	. . . . . . 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 2727	. . . . . 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 20180	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
94	12, 93	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Grp)
95	94	adantr 479	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ Grp)
96		eqid 2725	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
97	96	ringmgp 20181	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
98	12, 97	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘𝑅) ∈ Mnd)
99	98	adantr 479	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mnd)
100		mndmgm 18698	. . . . . . . . 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 20082	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
103	102, 83	mulgnncl 19046	. . . . . . . 8 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
104	101, 84, 81, 103	syl3anc 1368	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
105	31, 5, 89	grpsubeq0 18984	. . . . . . 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 1368	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋) = (0_g‘𝑅) ↔ (𝑁 ↑ 𝑦) = 𝑋))
107	92, 106	bitrd 278	. . . . 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 3426	. . . 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 2765	. . 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 6895	. 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 {crab 3419 Vcvv 3463 {csn 4624 class class class wbr 5143 ^◡ccnv 5671 “ cima 5675 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 0cc0 11136 ℝ^*cxr 11275 < clt 11276 ≤ cle 11277 ℕcn 12240 ℕ₀cn0 12500 ♯chash 14319 Basecbs 17177 0_gc0g 17418 ↑_s cpws 17425 Mgmcmgm 18595 Mndcmnd 18691 Grpcgrp 18892 -_gcsg 18894 ._gcmg 19025 mulGrpcmgp 20076 Ringcrg 20175 CRingccrg 20176 RingHom crh 20410 NzRingcnzr 20453 Domncdomn 21229 IDomncidom 21230 algSccascl 21788 var₁cv1 22101 Poly₁cpl1 22102 eval₁ce1 22240 deg₁ cdg1 26003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-rhm 20413 df-nzr 20454 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-rlreg 21232 df-domn 21233 df-idom 21234 df-cnfld 21282 df-assa 21789 df-asp 21790 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-evls 22023 df-evl 22024 df-psr1 22105 df-vr1 22106 df-ply1 22107 df-coe1 22108 df-evl1 22242 df-mdeg 26004 df-deg1 26005 df-mon1 26082 df-uc1p 26083 df-q1p 26084 df-r1p 26085

This theorem is referenced by: idomodle 42683

W3C validator