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Theorem aks5lem2 42954

Description: Lemma for section 5 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. Construct the quotient for the AKS reduction. (Contributed by metakunt, 7-Jun-2025.)

**Hypotheses**
Ref	Expression
aks5lem1.1	⊢ (𝜑 → 𝐾 ∈ Field)
aks5lem1.2	⊢ 𝑃 = (chr‘𝐾)
aks5lem1.3	⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
aks5lem1.4	⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))
aks5lem1.5	⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))
aks5lem1.6	⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀))
aks5lem2.1	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks5lem2.2	⊢ 𝐼 = (𝑠 ∈ (Base‘𝐴) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))
aks5lem2.3	⊢ 𝐴 = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
aks5lem2.4	⊢ 𝐿 = ((RSpan‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘{((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))})
aks5lem2.5	⊢ (𝜑 → 𝑅 ∈ ℕ)

**Assertion**
Ref	Expression
aks5lem2	⊢ (𝜑 → (𝐼 ∈ (𝐴 RingHom 𝐾) ∧ ∀𝑔 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑔]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑔)))

Distinct variable groups: 𝐴,𝑠 𝐹,𝑠 𝐺,𝑝 𝐻,𝑠 𝐼,𝑠 𝐾,𝑝 𝐾,𝑞 𝐾,𝑟 𝐾,𝑠 𝐿,𝑠 𝑀,𝑟 𝑁,𝑝 𝑁,𝑞 𝑁,𝑠 𝑅,𝑝 𝑅,𝑟 𝜑,𝑔,𝑠 𝜑,𝑝 𝜑,𝑟 Allowed substitution hints: 𝜑(𝑞) 𝐴(𝑔,𝑟,𝑞,𝑝) 𝑃(𝑔,𝑠,𝑟,𝑞,𝑝) 𝑅(𝑔,𝑠,𝑞) 𝐹(𝑔,𝑟,𝑞,𝑝) 𝐺(𝑔,𝑠,𝑟,𝑞) 𝐻(𝑔,𝑟,𝑞,𝑝) 𝐼(𝑔,𝑟,𝑞,𝑝) 𝐾(𝑔) 𝐿(𝑔,𝑟,𝑞,𝑝) 𝑀(𝑔,𝑠,𝑞,𝑝) 𝑁(𝑔,𝑟)

Proof of Theorem aks5lem2 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. 2 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
2		aks5lem1.1	. . 3 ⊢ (𝜑 → 𝐾 ∈ Field)
3		aks5lem1.2	. . 3 ⊢ 𝑃 = (chr‘𝐾)
4		aks5lem1.3	. . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
5		aks5lem1.4	. . 3 ⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))
6		aks5lem1.5	. . 3 ⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))
7		aks5lem1.6	. . 3 ⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀))
8		aks5lem2.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
9	2	fldcrngd 20842	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
10		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
11	10	crngmgp 20318	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
12	9, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
13		aks5lem2.5	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℕ)
14	13	nnnn0d 12560	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
15		eqid 2763	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
16	12, 14, 15	isprimroot 42860	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑙))))
17	8, 16	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑙)))
18	17	simp1d 1160	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
19		eqid 2763	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	10, 19	mgpbas 20216	. . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
21	20	eqcomi 2772	. . . 4 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝐾)
22	18, 21	eleqtrdi 2873	. . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
23	2, 3, 4, 5, 6, 7, 22	aks5lem1 42953	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
24		eqid 2763	. 2 ⊢ (^◡(𝐻 ∘ 𝐹) “ {(0_g‘𝐾)}) = (^◡(𝐻 ∘ 𝐹) “ {(0_g‘𝐾)})
25		aks5lem2.3	. 2 ⊢ 𝐴 = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
26		aks5lem2.2	. 2 ⊢ 𝐼 = (𝑠 ∈ (Base‘𝐴) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))
27	4	simp2d 1161	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28	27	nnnn0d 12560	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29		eqid 2763	. . . . 5 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
30	29	zncrng 21694	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
31	28, 30	syl 18	. . 3 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ CRing)
32		eqid 2763	. . . 4 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = (Poly₁‘(ℤ/nℤ‘𝑁))
33	32	ply1crng 22358	. . 3 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
34	31, 33	syl 18	. 2 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
35		aks5lem2.4	. 2 ⊢ 𝐿 = ((RSpan‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘{((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))})
36	34	crnggrpd 20324	. . 3 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
37		eqid 2763	. . . . 5 ⊢ (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
38		eqid 2763	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))
39	37, 38	mgpbas 20216	. . . 4 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
40		eqid 2763	. . . 4 ⊢ (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
41	34	crngringd 20323	. . . . 5 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring)
42	37	ringmgp 20316	. . . . 5 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
43	41, 42	syl 18	. . . 4 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
44	31	crngringd 20323	. . . . 5 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ Ring)
45		eqid 2763	. . . . . 6 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁))
46	45, 32, 38	vr1cl 22377	. . . . 5 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
47	44, 46	syl 18	. . . 4 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
48	39, 40, 43, 14, 47	mulgnn0cld 19156	. . 3 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
49		eqid 2763	. . . . 5 ⊢ (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
50	38, 49	ringidcl 20344	. . . 4 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
51	41, 50	syl 18	. . 3 ⊢ (𝜑 → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
52		eqid 2763	. . . 4 ⊢ (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
53	38, 52	grpsubcl 19081	. . 3 ⊢ (((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp ∧ (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∧ (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → ((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
54	36, 48, 51, 53	syl3anc 1398	. 2 ⊢ (𝜑 → ((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
55		fvexd 6896	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(ℤ/nℤ‘𝑁)) ∈ V)
56	55	mptexd 7222	. . . . . . . . 9 ⊢ (𝜑 → (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞)) ∈ V)
57	6, 56	eqeltrid 2867	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ V)
58	57	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → 𝐺 ∈ V)
59		vex 3459	. . . . . . . 8 ⊢ 𝑝 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → 𝑝 ∈ V)
61	58, 60	coexd 7924	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐺 ∘ 𝑝) ∈ V)
62	61, 5	fmptd 7109	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶V)
63	62	ffund 6710	. . . 4 ⊢ (𝜑 → Fun 𝐹)
64	62	fdmd 6716	. . . . 5 ⊢ (𝜑 → dom 𝐹 = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
65	54, 64	eleqtrrd 2866	. . . 4 ⊢ (𝜑 → ((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ dom 𝐹)
66		fvco 6979	. . . 4 ⊢ ((Fun 𝐹 ∧ ((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ dom 𝐹) → ((𝐻 ∘ 𝐹)‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))))
67	63, 65, 66	syl2anc 595	. . 3 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))))
68		eqid 2763	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
69	9	crngringd 20323	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
70	4	simp1d 1160	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
71		prmnn 16727	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
72	70, 71	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
73	3, 72	eqeltrrid 2868	. . . . . . . . . 10 ⊢ (𝜑 → (chr‘𝐾) ∈ ℕ)
74	73	nnzd 12612	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∈ ℤ)
75	4	simp3d 1162	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
76	3, 75	eqbrtrrid 5147	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁)
77	69, 27, 74, 76, 29, 6	zndvdchrrhm 42740	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾))
78	32, 68, 38, 5, 77	rhmply1 22543	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)))
79		rhmghm 20562	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
80	78, 79	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
81		eqid 2763	. . . . . . 7 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
82	38, 52, 81	ghmsub 19289	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∧ (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = ((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))))
83	80, 48, 51, 82	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = ((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))))
84	83	fveq2d 6885	. . . 4 ⊢ (𝜑 → (𝐻‘(𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = (𝐻‘((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))))
85		eqid 2763	. . . . . . . 8 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
86		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
87	85, 68, 19, 86, 9, 22, 7	evl1maprhm 22539	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾))
88		rhmghm 20562	. . . . . . 7 ⊢ (𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾) → 𝐻 ∈ ((Poly₁‘𝐾) GrpHom 𝐾))
89	87, 88	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ ((Poly₁‘𝐾) GrpHom 𝐾))
90	38, 86	rhmf 20563	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
91	78, 90	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
92	91, 48	ffvelcdmd 7080	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾)))
93	91, 51	ffvelcdmd 7080	. . . . . 6 ⊢ (𝜑 → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾)))
94		eqid 2763	. . . . . . 7 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
95	86, 81, 94	ghmsub 19289	. . . . . 6 ⊢ ((𝐻 ∈ ((Poly₁‘𝐾) GrpHom 𝐾) ∧ (𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾))) → (𝐻‘((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = ((𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))(-_g‘𝐾)(𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))))
96	89, 92, 93, 95	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝐻‘((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = ((𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))(-_g‘𝐾)(𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))))
97		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
98	38, 97, 49, 41, 48	ringlidmd 20351	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))
99	98	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) = ((1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
100	31	elexd 3478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ V)
101	32	ply1sca 22412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℤ/nℤ‘𝑁) ∈ V → (ℤ/nℤ‘𝑁) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
102	100, 101	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
103	102	fveq2d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
104	103	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁))) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))))
105		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))
106		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))
107	32	ply1lmod 22411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ LMod)
108	44, 107	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ LMod)
109	105, 106, 108, 41	ascl1 22035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))
110	104, 109	eqtrd 2798	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))
111	110	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁))))
112	111	oveq1d 7425	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = (((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
113	99, 112	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) = (((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
114	32	ply1assa 22359	. . . . . . . . . . . . . . . 16 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ AssAlg)
115	31, 114	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ AssAlg)
116		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
117		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
118	116, 117	ringidcl 20344	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)))
119	44, 118	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)))
120	102	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
121	119, 120	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
122		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
123		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁))) = ( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))
124	105, 106, 122, 38, 97, 123	asclmul1 22036	. . . . . . . . . . . . . . 15 ⊢ (((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ AssAlg ∧ (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∧ (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = ((1_r‘(ℤ/nℤ‘𝑁))( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
125	115, 121, 48, 124	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁)))(._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = ((1_r‘(ℤ/nℤ‘𝑁))( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
126	113, 125	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) = ((1_r‘(ℤ/nℤ‘𝑁))( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))
127	126	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = (𝐹‘((1_r‘(ℤ/nℤ‘𝑁))( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))))
128		eqid 2763	. . . . . . . . . . . . 13 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
129		eqid 2763	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐾)) = ( ·_𝑠 ‘(Poly₁‘𝐾))
130		eqid 2763	. . . . . . . . . . . . 13 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
131		eqid 2763	. . . . . . . . . . . . 13 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
132	32, 68, 38, 116, 5, 45, 128, 123, 129, 37, 130, 40, 131, 77, 119, 14	rhmply1mon 22546	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((1_r‘(ℤ/nℤ‘𝑁))( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = ((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
133	127, 132	eqtrd 2798	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = ((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
134	133	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))))
135		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
136	117, 135	rhm1 20572	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾) → (𝐺‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘𝐾))
137	77, 136	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘𝐾))
138	137	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = ((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
139	138	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻‘((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (𝐻‘((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))))
140	68	ply1assa 22359	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ AssAlg)
141	9, 140	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ AssAlg)
142	19, 135	ringidcl 20344	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (1_r‘𝐾) ∈ (Base‘𝐾))
143	69, 142	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘𝐾) ∈ (Base‘𝐾))
144	68	ply1sca 22412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Field → 𝐾 = (Scalar‘(Poly₁‘𝐾)))
145	2, 144	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 = (Scalar‘(Poly₁‘𝐾)))
146	145	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(Poly₁‘𝐾))))
147	143, 146	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝐾) ∈ (Base‘(Scalar‘(Poly₁‘𝐾))))
148	130, 86	mgpbas 20216	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
149	68	ply1crng 22358	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
150	9, 149	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
151		crngring 20322	. . . . . . . . . . . . . . . . . 18 ⊢ ((Poly₁‘𝐾) ∈ CRing → (Poly₁‘𝐾) ∈ Ring)
152	150, 151	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
153	130	ringmgp 20316	. . . . . . . . . . . . . . . . 17 ⊢ ((Poly₁‘𝐾) ∈ Ring → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
154	152, 153	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
155	128, 68, 86	vr1cl 22377	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
156	69, 155	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
157	148, 131, 154, 14, 156	mulgnn0cld 19156	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) ∈ (Base‘(Poly₁‘𝐾)))
158		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
159		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘(Poly₁‘𝐾)) = (Scalar‘(Poly₁‘𝐾))
160		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘(Poly₁‘𝐾))) = (Base‘(Scalar‘(Poly₁‘𝐾)))
161		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (._r‘(Poly₁‘𝐾)) = (._r‘(Poly₁‘𝐾))
162	158, 159, 160, 86, 161, 129	asclmul1 22036	. . . . . . . . . . . . . . 15 ⊢ (((Poly₁‘𝐾) ∈ AssAlg ∧ (1_r‘𝐾) ∈ (Base‘(Scalar‘(Poly₁‘𝐾))) ∧ (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) ∈ (Base‘(Poly₁‘𝐾))) → (((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = ((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
163	141, 147, 157, 162	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = ((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
164	163	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = (((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
165	164	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐻‘((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (𝐻‘(((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))))
166		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘(Poly₁‘𝐾)) = (1_r‘(Poly₁‘𝐾))
167	68, 158, 135, 166, 69	ply1ascl1 22415	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾)) = (1_r‘(Poly₁‘𝐾)))
168	167	oveq1d 7425	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = ((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
169	168	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻‘(((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (𝐻‘((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))))
170	86, 161, 166, 152, 157	ringlidmd 20351	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
171	170	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻‘((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (𝐻‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
172	7	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀)))
173		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) → 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
174	173	fveq2d 6885	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) → ((eval₁‘𝐾)‘𝑟) = ((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
175	174	fveq1d 6883	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) → (((eval₁‘𝐾)‘𝑟)‘𝑀) = (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀))
176		fvexd 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) ∈ V)
177	172, 175, 157, 176	fvmptd 6997	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀))
178	85, 128, 19, 68, 86, 9, 22	evl1vard 22497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(var₁‘𝐾))‘𝑀) = 𝑀))
179	85, 68, 19, 86, 9, 22, 178, 131, 15, 14	evl1expd 22505	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (𝑅(._g‘(mulGrp‘𝐾))𝑀)))
180	179	simprd 500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (𝑅(._g‘(mulGrp‘𝐾))𝑀))
181	17	simp2d 1161	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)))
182	10, 135	ringidval 20260	. . . . . . . . . . . . . . . . . . 19 ⊢ (1_r‘𝐾) = (0_g‘(mulGrp‘𝐾))
183	182	eqcomi 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘(mulGrp‘𝐾)) = (1_r‘𝐾)
184	183	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0_g‘(mulGrp‘𝐾)) = (1_r‘𝐾))
185	181, 184	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (1_r‘𝐾))
186	180, 185	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (1_r‘𝐾))
187	177, 186	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = (1_r‘𝐾))
188	171, 187	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻‘((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
189	169, 188	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐻‘(((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
190	165, 189	eqtrd 2798	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻‘((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
191	139, 190	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻‘((𝐺‘(1_r‘(ℤ/nℤ‘𝑁)))( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
192	134, 191	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
193	166, 135	rhm1 20572	. . . . . . . . . . 11 ⊢ (𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾) → (𝐻‘(1_r‘(Poly₁‘𝐾))) = (1_r‘𝐾))
194	87, 193	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻‘(1_r‘(Poly₁‘𝐾))) = (1_r‘𝐾))
195	194	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝐾) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
196	192, 195	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
197	196, 194	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
198	49, 166	rhm1 20572	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (1_r‘(Poly₁‘𝐾)))
199	78, 198	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (1_r‘(Poly₁‘𝐾)))
200	199	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
201	200, 194	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
202	197, 201	oveq12d 7428	. . . . . 6 ⊢ (𝜑 → ((𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))(-_g‘𝐾)(𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = ((1_r‘𝐾)(-_g‘𝐾)(1_r‘𝐾)))
203	69	ringgrpd 20319	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Grp)
204	19, 1, 94	grpsubid 19085	. . . . . . 7 ⊢ ((𝐾 ∈ Grp ∧ (1_r‘𝐾) ∈ (Base‘𝐾)) → ((1_r‘𝐾)(-_g‘𝐾)(1_r‘𝐾)) = (0_g‘𝐾))
205	203, 143, 204	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝐾)(-_g‘𝐾)(1_r‘𝐾)) = (0_g‘𝐾))
206	202, 205	eqtrd 2798	. . . . 5 ⊢ (𝜑 → ((𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))))(-_g‘𝐾)(𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = (0_g‘𝐾))
207	96, 206	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝐻‘((𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(-_g‘(Poly₁‘𝐾))(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = (0_g‘𝐾))
208	84, 207	eqtrd 2798	. . 3 ⊢ (𝜑 → (𝐻‘(𝐹‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))))) = (0_g‘𝐾))
209	67, 208	eqtrd 2798	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (0_g‘𝐾))
210	1, 23, 24, 25, 26, 34, 35, 54, 209	rhmqusspan 42952	1 ⊢ (𝜑 → (𝐼 ∈ (𝐴 RingHom 𝐾) ∧ ∀𝑔 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑔]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑔)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 {csn 4589 ∪ cuni 4872 class class class wbr 5109 ↦ cmpt 5192 ^◡ccnv 5660 dom cdm 5661 “ cima 5664 ∘ ccom 5665 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 [cec 8688 ℕcn 12228 ℕ₀cn0 12499 ∥ cdvds 16305 ℙcprime 16724 Basecbs 17264 ._rcmulr 17306 Scalarcsca 17308 ·_𝑠 cvsca 17309 0_gc0g 17487 /_s cqus 17554 Mndcmnd 18787 Grpcgrp 18995 -_gcsg 18997 ._gcmg 19128 ~_QG cqg 19183 GrpHom cghm 19278 CMndccmn 19845 mulGrpcmgp 20211 1_rcur 20258 Ringcrg 20310 CRingccrg 20311 RingHom crh 20547 Fieldcfield 20828 LModclmod 20981 RSpancrsp 21331 ℤRHomczrh 21649 chrcchr 21651 ℤ/nℤczn 21652 AssAlgcasa 22000 algSccascl 22002 var₁cv1 22336 Poly₁cpl1 22337 eval₁ce1 22474 PrimRoots cprimroots 42858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-prm 16725 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-0g 17489 df-gsum 17490 df-prds 17495 df-pws 17497 df-imas 17557 df-qus 17558 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-nsg 19185 df-eqg 19186 df-ghm 19279 df-cntz 19382 df-od 19593 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-srg 20264 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-field 20830 df-lmod 20983 df-lss 21053 df-lsp 21093 df-sra 21294 df-rgmod 21295 df-lidl 21332 df-rsp 21333 df-2idl 21389 df-cnfld 21523 df-zring 21597 df-zrh 21653 df-chr 21655 df-zn 21656 df-assa 22003 df-asp 22004 df-ascl 22005 df-psr 22059 df-mvr 22060 df-mpl 22061 df-opsr 22063 df-evls 22225 df-evl 22226 df-psr1 22340 df-vr1 22341 df-ply1 22342 df-coe1 22343 df-evls1 22475 df-evl1 22476 df-primroots 42859

This theorem is referenced by: aks5lem3a 42956

W3C validator