idomrootle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > idomrootle		Structured version Visualization version GIF version

Theorem idomrootle 26521

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 2763	. . 3 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
2		eqid 2763	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
3		eqid 2763	. . 3 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
4		eqid 2763	. . 3 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
5		eqid 2763	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
6		eqid 2763	. . 3 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
7		simp1 1154	. . 3 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ IDomn)
8		isidom 21006	. . . . . . . . 9 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
9	8	simplbi 502	. . . . . . . 8 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
10	7, 9	syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ CRing)
11		crngring 20502	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Ring)
13	1	ply1ring 22595	. . . . . 6 ⊢ (𝑅 ∈ Ring → (Poly₁‘𝑅) ∈ Ring)
14	12, 13	syl 18	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Ring)
15		ringgrp 20494	. . . . 5 ⊢ ((Poly₁‘𝑅) ∈ Ring → (Poly₁‘𝑅) ∈ Grp)
16	14, 15	syl 18	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (Poly₁‘𝑅) ∈ Grp)
17		eqid 2763	. . . . . . . 8 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
18	17	ringmgp 20495	. . . . . . 7 ⊢ ((Poly₁‘𝑅) ∈ Ring → (mulGrp‘(Poly₁‘𝑅)) ∈ Mnd)
19	14, 18	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘(Poly₁‘𝑅)) ∈ Mnd)
20		mndmgm 18959	. . . . . 6 ⊢ ((mulGrp‘(Poly₁‘𝑅)) ∈ Mnd → (mulGrp‘(Poly₁‘𝑅)) ∈ Mgm)
21	19, 20	syl 18	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘(Poly₁‘𝑅)) ∈ Mgm)
22		simp3 1156	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
23		eqid 2763	. . . . . . 7 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24	23, 1, 2	vr1cl 22565	. . . . . 6 ⊢ (𝑅 ∈ Ring → (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅)))
25	12, 24	syl 18	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (var₁‘𝑅) ∈ (Base‘(Poly₁‘𝑅)))
26	17, 2	mgpbas 20395	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
27		eqid 2763	. . . . . 6 ⊢ (._g‘(mulGrp‘(Poly₁‘𝑅))) = (._g‘(mulGrp‘(Poly₁‘𝑅)))
28	26, 27	mulgnncl 19328	. . . . 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 1399	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅)) ∈ (Base‘(Poly₁‘𝑅)))
30		eqid 2763	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
31		idomrootle.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
32	1, 30, 31, 2	ply1sclf 22634	. . . . . 6 ⊢ (𝑅 ∈ Ring → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(Poly₁‘𝑅)))
33	12, 32	syl 18	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(Poly₁‘𝑅)))
34		simp2 1155	. . . . 5 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵)
35	33, 34	ffvelcdmd 7109	. . . 4 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)))
36		eqid 2763	. . . . 5 ⊢ (-_g‘(Poly₁‘𝑅)) = (-_g‘(Poly₁‘𝑅))
37	2, 36	grpsubcl 19259	. . . 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 1399	. . 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 26430	. . . . . . . . . 10 ⊢ (((algSc‘(Poly₁‘𝑅))‘𝑋) ∈ (Base‘(Poly₁‘𝑅)) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ ℝ^*)
40	35, 39	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ∈ ℝ^*)
41		0xr 11383	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
42	41	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ^*)
43		nnre 12369	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
44	43	rexrd 11386	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)
45	44	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ^*)
46	3, 1, 31, 30	deg1sclle 26460	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
47	12, 34, 46	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) ≤ 0)
48		nngt0 12396	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
49	48	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
50	40, 42, 45, 47, 49	xrlelttrd 13316	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑋)) < 𝑁)
51	8	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Domn)
52		domnnzr 20988	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
53	51, 52	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ NzRing)
54	7, 53	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ NzRing)
55		nnnn0 12640	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
56	55	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
57	3, 1, 23, 17, 27	deg1pw 26469	. . . . . . . . 9 ⊢ ((𝑅 ∈ NzRing ∧ 𝑁 ∈ ℕ₀) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
58	54, 56, 57	syl2anc 596	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → ((deg₁‘𝑅)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝑅)))(var₁‘𝑅))) = 𝑁)
59	50, 58	breqtrrd 5136	. . . . . . 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 26456	. . . . . 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 2798	. . . . 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 2864	. . . 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 26438	. . . . 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 596	. . . 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 260	. . 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 26518	. 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 2763	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
68		eqid 2763	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
69	31	fvexi 6922	. . . . . . . 8 ⊢ 𝐵 ∈ V
70	69	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ V)
71	4, 1, 67, 31	evl1rhm 22680	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
72	10, 71	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
73	2, 68	rhmf 20745	. . . . . . . . 9 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
74	72, 73	syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
75	74, 38	ffvelcdmd 7109	. . . . . . 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 17689	. . . . . 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 6732	. . . . 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 7085	. . . . 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 18	. . . 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 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
81		simpr 490	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
82	4, 23, 31, 1, 2, 80, 81	evl1vard 22685	. . . . . . . . . 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 1212	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑁 ∈ ℕ)
85	84, 55	syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑁 ∈ ℕ₀)
86	4, 1, 31, 2, 80, 81, 82, 27, 83, 85	evl1expd 22693	. . . . . . . . 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 1211	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
88	4, 1, 31, 30, 2, 80, 87, 81	evl1scad 22683	. . . . . . . . 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 2763	. . . . . . . . 9 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
90	4, 1, 31, 2, 80, 81, 86, 88, 36, 89	evl1subd 22690	. . . . . . . 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 501	. . . . . . 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 2765	. . . . . 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 20494	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
94	12, 93	syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → 𝑅 ∈ Grp)
95	94	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ Grp)
96		eqid 2763	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
97	96	ringmgp 20495	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
98	12, 97	syl 18	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (mulGrp‘𝑅) ∈ Mnd)
99	98	adantr 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mnd)
100		mndmgm 18959	. . . . . . . . 9 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm)
101	99, 100	syl 18	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mgm)
102	96, 31	mgpbas 20395	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
103	102, 83	mulgnncl 19328	. . . . . . . 8 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
104	101, 84, 81, 103	syl3anc 1399	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (𝑁 ↑ 𝑦) ∈ 𝐵)
105	31, 5, 89	grpsubeq0 19265	. . . . . . 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 1399	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) ∧ 𝑦 ∈ 𝐵) → (((𝑁 ↑ 𝑦)(-_g‘𝑅)𝑋) = (0_g‘𝑅) ↔ (𝑁 ↑ 𝑦) = 𝑋))
107	92, 106	bitrd 282	. . . . 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 3422	. . . 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 2798	. . 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 6912	. 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 5139	1 ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≤ 𝑁)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 {crab 3416 Vcvv 3454 {csn 4588 class class class wbr 5106 ^◡ccnv 5654 “ cima 5658 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 0cc0 11227 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 ℕcn 12362 ℕ₀cn0 12633 ♯chash 14501 Basecbs 17416 0_gc0g 17639 ↑_s cpws 17646 Mgmcmgm 18843 Mndcmnd 18952 Grpcgrp 19173 -_gcsg 19175 ._gcmg 19306 mulGrpcmgp 20390 Ringcrg 20489 CRingccrg 20490 RingHom crh 20729 NzRingcnzr 20792 Domncdomn 20974 IDomncidom 20975 algSccascl 22190 var₁cv1 22524 Poly₁cpl1 22525 eval₁ce1 22662 deg₁cdg1 26402

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 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 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-ofr 7718 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-oi 9531 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-xnn0 12707 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-fzo 13816 df-seq 14172 df-hash 14502 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-0g 17641 df-gsum 17642 df-prds 17647 df-pws 17649 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-srg 20443 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-rhm 20732 df-nzr 20793 df-subrng 20828 df-subrg 20852 df-rlreg 20976 df-domn 20977 df-idom 20978 df-lmod 21167 df-lss 21237 df-lsp 21277 df-cnfld 21709 df-assa 22191 df-asp 22192 df-ascl 22193 df-psr 22247 df-mvr 22248 df-mpl 22249 df-opsr 22251 df-evls 22413 df-evl 22414 df-psr1 22528 df-vr1 22529 df-ply1 22530 df-coe1 22531 df-evl1 22664 df-mdeg 26403 df-deg1 26404 df-mon1 26479 df-uc1p 26480 df-q1p 26481 df-r1p 26482

This theorem is used by: unitscyglem5 43441 idomodle 44385

W3C validator