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Theorem aks5lem2 42626

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 2736	. 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 20719	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
10		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
11	10	crngmgp 20222	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
13		aks5lem2.5	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℕ)
14	13	nnnn0d 12498	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
15		eqid 2736	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
16	12, 14, 15	isprimroot 42532	. . . . . 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 232	. . . . 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 1143	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
19		eqid 2736	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	10, 19	mgpbas 20126	. . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
21	20	eqcomi 2745	. . . 4 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝐾)
22	18, 21	eleqtrdi 2846	. . 3 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
23	2, 3, 4, 5, 6, 7, 22	aks5lem1 42625	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
24		eqid 2736	. 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 1144	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28	27	nnnn0d 12498	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29		eqid 2736	. . . . 5 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
30	29	zncrng 21524	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
31	28, 30	syl 17	. . 3 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ CRing)
32		eqid 2736	. . . 4 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = (Poly₁‘(ℤ/nℤ‘𝑁))
33	32	ply1crng 22162	. . 3 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
34	31, 33	syl 17	. 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 20228	. . 3 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
37		eqid 2736	. . . . 5 ⊢ (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
38		eqid 2736	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))
39	37, 38	mgpbas 20126	. . . 4 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
40		eqid 2736	. . . 4 ⊢ (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
41	34	crngringd 20227	. . . . 5 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring)
42	37	ringmgp 20220	. . . . 5 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
43	41, 42	syl 17	. . . 4 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
44	31	crngringd 20227	. . . . 5 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ Ring)
45		eqid 2736	. . . . . 6 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁))
46	45, 32, 38	vr1cl 22181	. . . . 5 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
47	44, 46	syl 17	. . . 4 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
48	39, 40, 43, 14, 47	mulgnn0cld 19071	. . 3 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
49		eqid 2736	. . . . 5 ⊢ (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
50	38, 49	ringidcl 20246	. . . 4 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
51	41, 50	syl 17	. . 3 ⊢ (𝜑 → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
52		eqid 2736	. . . 4 ⊢ (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
53	38, 52	grpsubcl 18996	. . 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 1374	. 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 6855	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(ℤ/nℤ‘𝑁)) ∈ V)
56	55	mptexd 7179	. . . . . . . . 9 ⊢ (𝜑 → (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞)) ∈ V)
57	6, 56	eqeltrid 2840	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ V)
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → 𝐺 ∈ V)
59		vex 3433	. . . . . . . 8 ⊢ 𝑝 ∈ V
60	59	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → 𝑝 ∈ V)
61	58, 60	coexd 7882	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐺 ∘ 𝑝) ∈ V)
62	61, 5	fmptd 7066	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶V)
63	62	ffund 6672	. . . 4 ⊢ (𝜑 → Fun 𝐹)
64	62	fdmd 6678	. . . . 5 ⊢ (𝜑 → dom 𝐹 = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
65	54, 64	eleqtrrd 2839	. . . 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 6938	. . . 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 585	. . 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 2736	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
69	9	crngringd 20227	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
70	4	simp1d 1143	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
71		prmnn 16643	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
72	70, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
73	3, 72	eqeltrrid 2841	. . . . . . . . . 10 ⊢ (𝜑 → (chr‘𝐾) ∈ ℕ)
74	73	nnzd 12550	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∈ ℤ)
75	4	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
76	3, 75	eqbrtrrid 5121	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁)
77	69, 27, 74, 76, 29, 6	zndvdchrrhm 42412	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾))
78	32, 68, 38, 5, 77	rhmply1 22351	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)))
79		rhmghm 20463	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
80	78, 79	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
81		eqid 2736	. . . . . . 7 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
82	38, 52, 81	ghmsub 19199	. . . . . 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 1374	. . . . 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 6844	. . . 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 2736	. . . . . . . 8 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
86		eqid 2736	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
87	85, 68, 19, 86, 9, 22, 7	evl1maprhm 22344	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾))
88		rhmghm 20463	. . . . . . 7 ⊢ (𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾) → 𝐻 ∈ ((Poly₁‘𝐾) GrpHom 𝐾))
89	87, 88	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ ((Poly₁‘𝐾) GrpHom 𝐾))
90	38, 86	rhmf 20464	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
91	78, 90	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
92	91, 48	ffvelcdmd 7037	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾)))
93	91, 51	ffvelcdmd 7037	. . . . . 6 ⊢ (𝜑 → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) ∈ (Base‘(Poly₁‘𝐾)))
94		eqid 2736	. . . . . . 7 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
95	86, 81, 94	ghmsub 19199	. . . . . 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 1374	. . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (._r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
98	38, 97, 49, 41, 48	ringlidmd 20253	. . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . 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 3453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ V)
101	32	ply1sca 22216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℤ/nℤ‘𝑁) ∈ V → (ℤ/nℤ‘𝑁) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
103	102	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
104	103	fveq2d 6844	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))
106		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))
107	32	ply1lmod 22215	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ LMod)
108	44, 107	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ LMod)
109	105, 106, 108, 41	ascl1 21865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))
110	104, 109	eqtrd 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))
111	110	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘(1_r‘(ℤ/nℤ‘𝑁))))
112	111	oveq1d 7382	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . 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 22163	. . . . . . . . . . . . . . . 16 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ AssAlg)
115	31, 114	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ AssAlg)
116		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
117		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
118	116, 117	ringidcl 20246	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)))
119	44, 118	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(ℤ/nℤ‘𝑁)))
120	102	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
121	119, 120	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
122		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (Base‘(Scalar‘(Poly₁‘(ℤ/nℤ‘𝑁))))
123		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁))) = ( ·_𝑠 ‘(Poly₁‘(ℤ/nℤ‘𝑁)))
124	105, 106, 122, 38, 97, 123	asclmul1 21866	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 6844	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
129		eqid 2736	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐾)) = ( ·_𝑠 ‘(Poly₁‘𝐾))
130		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
131		eqid 2736	. . . . . . . . . . . . 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 22354	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 6844	. . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
136	117, 135	rhm1 20468	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾) → (𝐺‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘𝐾))
137	77, 136	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘(1_r‘(ℤ/nℤ‘𝑁))) = (1_r‘𝐾))
138	137	oveq1d 7382	. . . . . . . . . . . 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 6844	. . . . . . . . . . 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 22163	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ AssAlg)
141	9, 140	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ AssAlg)
142	19, 135	ringidcl 20246	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (1_r‘𝐾) ∈ (Base‘𝐾))
143	69, 142	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1_r‘𝐾) ∈ (Base‘𝐾))
144	68	ply1sca 22216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Field → 𝐾 = (Scalar‘(Poly₁‘𝐾)))
145	2, 144	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 = (Scalar‘(Poly₁‘𝐾)))
146	145	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(Poly₁‘𝐾))))
147	143, 146	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝐾) ∈ (Base‘(Scalar‘(Poly₁‘𝐾))))
148	130, 86	mgpbas 20126	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
149	68	ply1crng 22162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
150	9, 149	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
151		crngring 20226	. . . . . . . . . . . . . . . . . 18 ⊢ ((Poly₁‘𝐾) ∈ CRing → (Poly₁‘𝐾) ∈ Ring)
152	150, 151	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
153	130	ringmgp 20220	. . . . . . . . . . . . . . . . 17 ⊢ ((Poly₁‘𝐾) ∈ Ring → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
154	152, 153	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
155	128, 68, 86	vr1cl 22181	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
156	69, 155	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
157	148, 131, 154, 14, 156	mulgnn0cld 19071	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) ∈ (Base‘(Poly₁‘𝐾)))
158		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
159		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘(Poly₁‘𝐾)) = (Scalar‘(Poly₁‘𝐾))
160		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Scalar‘(Poly₁‘𝐾))) = (Base‘(Scalar‘(Poly₁‘𝐾)))
161		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (._r‘(Poly₁‘𝐾)) = (._r‘(Poly₁‘𝐾))
162	158, 159, 160, 86, 161, 129	asclmul1 21866	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . 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 6844	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘(Poly₁‘𝐾)) = (1_r‘(Poly₁‘𝐾))
167	68, 158, 135, 166, 69	ply1ascl1 22219	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝐾))‘(1_r‘𝐾)) = (1_r‘(Poly₁‘𝐾)))
168	167	oveq1d 7382	. . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . 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 20253	. . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) → 𝑟 = (𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
174	173	fveq2d 6844	. . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) ∈ V)
177	172, 175, 157, 176	fvmptd 6955	. . . . . . . . . . . . . . 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 22302	. . . . . . . . . . . . . . . . . 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 22310	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (𝑅(._g‘(mulGrp‘𝐾))𝑀))
181	17	simp2d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)))
182	10, 135	ringidval 20164	. . . . . . . . . . . . . . . . . . 19 ⊢ (1_r‘𝐾) = (0_g‘(mulGrp‘𝐾))
183	182	eqcomi 2745	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (1_r‘𝐾))
186	180, 185	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (1_r‘𝐾))
187	177, 186	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻‘(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = (1_r‘𝐾))
188	171, 187	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐻‘((1_r‘(Poly₁‘𝐾))(._r‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
189	169, 188	eqtrd 2771	. . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻‘((1_r‘𝐾)( ·_𝑠 ‘(Poly₁‘𝐾))(𝑅(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))) = (1_r‘𝐾))
191	139, 190	eqtrd 2771	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
193	166, 135	rhm1 20468	. . . . . . . . . . 11 ⊢ (𝐻 ∈ ((Poly₁‘𝐾) RingHom 𝐾) → (𝐻‘(1_r‘(Poly₁‘𝐾))) = (1_r‘𝐾))
194	87, 193	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻‘(1_r‘(Poly₁‘𝐾))) = (1_r‘𝐾))
195	194	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝐾) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
196	192, 195	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
197	196, 194	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(𝐹‘(𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
198	49, 166	rhm1 20468	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (1_r‘(Poly₁‘𝐾)))
199	78, 198	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (1_r‘(Poly₁‘𝐾)))
200	199	fveq2d 6844	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (𝐻‘(1_r‘(Poly₁‘𝐾))))
201	200, 194	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(𝐹‘(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))) = (1_r‘𝐾))
202	197, 201	oveq12d 7385	. . . . . 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 20223	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Grp)
204	19, 1, 94	grpsubid 19000	. . . . . . 7 ⊢ ((𝐾 ∈ Grp ∧ (1_r‘𝐾) ∈ (Base‘𝐾)) → ((1_r‘𝐾)(-_g‘𝐾)(1_r‘𝐾)) = (0_g‘𝐾))
205	203, 143, 204	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝐾)(-_g‘𝐾)(1_r‘𝐾)) = (0_g‘𝐾))
206	202, 205	eqtrd 2771	. . . . 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 2771	. . . 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 2771	. . 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 2771	. 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 42624	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 {csn 4567 ∪ cuni 4850 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 dom cdm 5631 “ cima 5634 ∘ ccom 5635 Fun wfun 6492 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 [cec 8641 ℕcn 12174 ℕ₀cn0 12437 ∥ cdvds 16221 ℙcprime 16640 Basecbs 17179 ._rcmulr 17221 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 /_s cqus 17469 Mndcmnd 18702 Grpcgrp 18909 -_gcsg 18911 ._gcmg 19043 ~_QG cqg 19098 GrpHom cghm 19187 CMndccmn 19755 mulGrpcmgp 20121 1_rcur 20162 Ringcrg 20214 CRingccrg 20215 RingHom crh 20449 Fieldcfield 20707 LModclmod 20855 RSpancrsp 21205 ℤRHomczrh 21479 chrcchr 21481 ℤ/nℤczn 21482 AssAlgcasa 21830 algSccascl 21832 var₁cv1 22139 Poly₁cpl1 22140 eval₁ce1 22279 PrimRoots cprimroots 42530

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-dvds 16222 df-prm 16641 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-imas 17472 df-qus 17473 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cntz 19292 df-od 19503 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-srg 20168 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-field 20709 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-chr 21485 df-zn 21486 df-assa 21833 df-asp 21834 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-evls 22052 df-evl 22053 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-evls1 22280 df-evl1 22281 df-primroots 42531

This theorem is referenced by: aks5lem3a 42628

W3C validator