aks5lem3a - Mathbox for metakunt

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Theorem aks5lem3a 41898

Description: Lemma for AKS section 5. (Contributed by metakunt, 17-Jun-2025.)

**Hypotheses**
Ref	Expression
aks5lema.1	⊢ (𝜑 → 𝐾 ∈ Field)
aks5lema.2	⊢ 𝑃 = (chr‘𝐾)
aks5lema.3	⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
aks5lema.9	⊢ 𝐵 = (𝑆 /_s (𝑆 ~_QG 𝐿))
aks5lema.10	⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))})
aks5lema.11	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks5lema.14	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
aks5lema.15	⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
aks5lem3a.4	⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))
aks5lem3a.5	⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))
aks5lem3a.6	⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀))
aks5lem3a.7	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks5lem3a.8	⊢ 𝐼 = (𝑠 ∈ (Base‘𝐵) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))
aks5lem3a.12	⊢ (𝜑 → 𝐴 ∈ ℤ)
aks5lem3a.13	⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿))

**Assertion**
Ref	Expression
aks5lem3a	⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))

Distinct variable groups: 𝐴,𝑟 𝐴,𝑠 𝐹,𝑟 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,𝑟 𝑁,𝑠 𝑅,𝑝 𝑅,𝑟 𝜑,𝑝 𝜑,𝑟 𝜑,𝑠 𝐵,𝑠 Allowed substitution hints: 𝜑(𝑦,𝑒,𝑓,𝑞) 𝐴(𝑦,𝑒,𝑓,𝑞,𝑝) 𝐵(𝑦,𝑒,𝑓,𝑟,𝑞,𝑝) 𝑃(𝑦,𝑒,𝑓,𝑠,𝑟,𝑞,𝑝) ∼ (𝑦,𝑒,𝑓,𝑠,𝑟,𝑞,𝑝) 𝑅(𝑦,𝑒,𝑓,𝑠,𝑞) 𝑆(𝑦,𝑒,𝑓,𝑠,𝑟,𝑞,𝑝) 𝐹(𝑦,𝑒,𝑓,𝑞,𝑝) 𝐺(𝑦,𝑒,𝑓,𝑠,𝑟,𝑞) 𝐻(𝑦,𝑒,𝑓,𝑟,𝑞,𝑝) 𝐼(𝑦,𝑒,𝑓,𝑟,𝑞,𝑝) 𝐾(𝑦,𝑒,𝑓) 𝐿(𝑦,𝑒,𝑓,𝑟,𝑞,𝑝) 𝑀(𝑦,𝑒,𝑓,𝑠,𝑞,𝑝) 𝑁(𝑦,𝑒,𝑓)

Proof of Theorem aks5lem3a Dummy variables 𝑢 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks5lema.1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Field)
2		aks5lema.2	. . . . . 6 ⊢ 𝑃 = (chr‘𝐾)
3		aks5lema.3	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
4		aks5lem3a.4	. . . . . 6 ⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))
5		aks5lem3a.5	. . . . . 6 ⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))
6		aks5lem3a.6	. . . . . 6 ⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀))
7		aks5lem3a.7	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
8	1	fldcrngd 20713	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
9		eqid 2726	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
10	9	crngmgp 20217	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
11	8, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
12		aks5lema.11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12575	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		eqid 2726	. . . . . . . . . 10 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
15	11, 13, 14	isprimroot 41802	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑑 ∈ ℕ₀ ((𝑑(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑑))))
16	7, 15	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑑 ∈ ℕ₀ ((𝑑(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑑)))
17	16	simp1d 1139	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
18		eqid 2726	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	9, 18	mgpbas 20116	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
20	19	eqcomi 2735	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝐾)
21	17, 20	eleqtrdi 2836	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
22	1, 2, 3, 4, 5, 6, 21	aks5lem1 41895	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
23		eqid 2726	. . . . . 6 ⊢ (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
24	23, 9	rhmmhm 20454	. . . . 5 ⊢ ((𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾) → (𝐻 ∘ 𝐹) ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘𝐾)))
25	22, 24	syl 17	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘𝐾)))
26	3	simp2d 1140	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12575	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		eqid 2726	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))
29		eqid 2726	. . . . 5 ⊢ (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
30		eqid 2726	. . . . . . . . . 10 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
31	30	zncrng 21535	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ CRing)
33		eqid 2726	. . . . . . . . 9 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = (Poly₁‘(ℤ/nℤ‘𝑁))
34	33	ply1crng 22181	. . . . . . . 8 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
35	32, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
36	35	crngringd 20222	. . . . . 6 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring)
37		ringgrp 20214	. . . . . 6 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
39	32	crngringd 20222	. . . . . 6 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ Ring)
40		eqid 2726	. . . . . . 7 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁))
41	40, 33, 28	vr1cl 22200	. . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
42	39, 41	syl 17	. . . . 5 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
43		eqid 2726	. . . . . . 7 ⊢ (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))
44		eqid 2726	. . . . . . 7 ⊢ (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))
45		eqid 2726	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
46		aks5lem3a.12	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
47	33, 43, 44, 45, 32, 46	ply1asclzrhval 41897	. . . . . 6 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))
48	44	zrhrhm 21494	. . . . . . . . 9 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (ℤ_ring RingHom (Poly₁‘(ℤ/nℤ‘𝑁))))
49		zringbas 21436	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
50	49, 28	rhmf 20460	. . . . . . . . 9 ⊢ ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (ℤ_ring RingHom (Poly₁‘(ℤ/nℤ‘𝑁))) → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))):ℤ⟶(Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
51	48, 50	syl 17	. . . . . . . 8 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))):ℤ⟶(Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
52	36, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))):ℤ⟶(Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
53	52, 46	ffvelcdmd 7088	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
54	47, 53	eqeltrd 2826	. . . . 5 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
55	28, 29, 38, 42, 54	grpcld 18934	. . . 4 ⊢ (𝜑 → ((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
56	23, 28	mgpbas 20116	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
57		eqid 2726	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
58	56, 57, 14	mhmmulg 19102	. . . 4 ⊢ (((𝐻 ∘ 𝐹) ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘𝐾)) ∧ 𝑁 ∈ ℕ₀ ∧ ((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (𝑁(._g‘(mulGrp‘𝐾))((𝐻 ∘ 𝐹)‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
59	25, 27, 55, 58	syl3anc 1368	. . 3 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (𝑁(._g‘(mulGrp‘𝐾))((𝐻 ∘ 𝐹)‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
60		eqid 2726	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
61	8	crngringd 20222	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
62	2	eqcomi 2735	. . . . . . . . . 10 ⊢ (chr‘𝐾) = 𝑃
63	3	simp1d 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
64		prmnn 16667	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
65	63, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
66	65	nnzd 12628	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	62, 66	eqeltrid 2830	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∈ ℤ)
68	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (chr‘𝐾) = 𝑃)
69	3	simp3d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
70	68, 69	eqbrtrd 5165	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁)
71	61, 26, 67, 70, 30, 5	zndvdchrrhm 41679	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾))
72	33, 60, 28, 4, 71	rhmply1 22371	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)))
73		eqid 2726	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
74	28, 73	rhmf 20460	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
75	72, 74	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
76	75, 55	fvco3d 6991	. . . . 5 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = (𝐻‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
77	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (𝑟 ∈ (Base‘(Poly₁‘𝐾)) ↦ (((eval₁‘𝐾)‘𝑟)‘𝑀)))
78		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → 𝑟 = (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
79	78	fveq2d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → ((eval₁‘𝐾)‘𝑟) = ((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
80	79	fveq1d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → (((eval₁‘𝐾)‘𝑟)‘𝑀) = (((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀))
81	75, 55	ffvelcdmd 7088	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) ∈ (Base‘(Poly₁‘𝐾)))
82		fvexd 6905	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) ∈ V)
83	77, 80, 81, 82	fvmptd 7005	. . . . . 6 ⊢ (𝜑 → (𝐻‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀))
84		rhmghm 20459	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
85	72, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
86		eqid 2726	. . . . . . . . . . 11 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
87	28, 29, 86	ghmlin 19208	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)) ∧ (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∧ ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
88	85, 42, 54, 87	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
89		eqid 2726	. . . . . . . . . . 11 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
90	33, 60, 28, 4, 40, 89, 71	rhmply1vr1 22372	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(var₁‘(ℤ/nℤ‘𝑁))) = (var₁‘𝐾))
91	47	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)))
92		eqid 2726	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘(Poly₁‘𝐾)) = (ℤRHom‘(Poly₁‘𝐾))
93	72, 46, 44, 92	rhmzrhval 41678	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
94	91, 93	eqtrd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
95		eqid 2726	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
96		eqid 2726	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
97	60, 95, 92, 96, 8, 46	ply1asclzrhval 41897	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
98	94, 97	eqtr4d 2769	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))
99	90, 98	oveq12d 7431	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))
100	88, 99	eqtrd 2766	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))
101	100	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → ((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = ((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))))
102	101	fveq1d 6892	. . . . . 6 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀))
103	83, 102	eqtrd 2766	. . . . 5 ⊢ (𝜑 → (𝐻‘(𝐹‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀))
104	76, 103	eqtrd 2766	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀))
105	104	oveq2d 7429	. . 3 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))((𝐻 ∘ 𝐹)‘((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (𝑁(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)))
106	59, 105	eqtr2d 2767	. 2 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
107		eceq1 8762	. . . . . . 7 ⊢ (𝑢 = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) → [𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
108	107	fveq2d 6894	. . . . . 6 ⊢ (𝑢 = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) → (𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = (𝐼‘[(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)))
109		fveq2 6890	. . . . . 6 ⊢ (𝑢 = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) → ((𝐻 ∘ 𝐹)‘𝑢) = ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
110	108, 109	eqeq12d 2742	. . . . 5 ⊢ (𝑢 = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) → ((𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢) ↔ (𝐼‘[(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))))
111		aks5lem3a.8	. . . . . . 7 ⊢ 𝐼 = (𝑠 ∈ (Base‘𝐵) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))
112		aks5lema.9	. . . . . . . 8 ⊢ 𝐵 = (𝑆 /_s (𝑆 ~_QG 𝐿))
113		aks5lema.15	. . . . . . . . 9 ⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
114	113	oveq1i 7423	. . . . . . . . 9 ⊢ (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)
115	113, 114	oveq12i 7425	. . . . . . . 8 ⊢ (𝑆 /_s (𝑆 ~_QG 𝐿)) = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
116	112, 115	eqtri 2754	. . . . . . 7 ⊢ 𝐵 = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
117		aks5lema.10	. . . . . . . 8 ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))})
118	113	fveq2i 6893	. . . . . . . . 9 ⊢ (RSpan‘𝑆) = (RSpan‘(Poly₁‘(ℤ/nℤ‘𝑁)))
119	113	fveq2i 6893	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
120	119	fveq2i 6893	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
121	120	oveqi 7426	. . . . . . . . . . 11 ⊢ (𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))
122	113	fveq2i 6893	. . . . . . . . . . 11 ⊢ (1_r‘𝑆) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
123	113	fveq2i 6893	. . . . . . . . . . 11 ⊢ (-_g‘𝑆) = (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
124	121, 122, 123	oveq123i 7427	. . . . . . . . . 10 ⊢ ((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆)) = ((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))
125	124	sneqi 4634	. . . . . . . . 9 ⊢ {((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))} = {((𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))(1_r‘(Poly₁‘(ℤ/nℤ‘𝑁))))}
126	118, 125	fveq12i 6896	. . . . . . . 8 ⊢ ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))}) = ((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))))})
127	117, 126	eqtri 2754	. . . . . . 7 ⊢ 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))))})
128	1, 2, 3, 4, 5, 6, 7, 111, 116, 127, 12	aks5lem2 41896	. . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (𝐵 RingHom 𝐾) ∧ ∀𝑢 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢)))
129	128	simprd 494	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢))
130	23	ringmgp 20215	. . . . . . 7 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
131	36, 130	syl 17	. . . . . 6 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
132	56, 57, 131, 27, 55	mulgnn0cld 19082	. . . . 5 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
133	110, 129, 132	rspcdva 3608	. . . 4 ⊢ (𝜑 → (𝐼‘[(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
134	133	eqcomd 2732	. . 3 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (𝐼‘[(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)))
135	113	eqcomi 2735	. . . . . . . . . . 11 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆
136	135	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆)
137	136	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘𝑆))
138	137	fveq2d 6894	. . . . . . . 8 ⊢ (𝜑 → (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘𝑆)))
139		eqidd 2727	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = 𝑁)
140	136	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘𝑆))
141		eqidd 2727	. . . . . . . . 9 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁)))
142	136	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘𝑆))
143	142	fveq1d 6892	. . . . . . . . 9 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
144	140, 141, 143	oveq123d 7434	. . . . . . . 8 ⊢ (𝜑 → ((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))
145	138, 139, 144	oveq123d 7434	. . . . . . 7 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = (𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
146	145	eceq1d 8763	. . . . . 6 ⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
147	136	oveq1d 7428	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = (𝑆 ~_QG 𝐿))
148	147	eceq2d 8766	. . . . . 6 ⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~_QG 𝐿))
149	146, 148	eqtrd 2766	. . . . 5 ⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~_QG 𝐿))
150		aks5lem3a.13	. . . . 5 ⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿))
151		eqcom 2733	. . . . . . . . . . 11 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆 ↔ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
152	151	imbi2i 335	. . . . . . . . . 10 ⊢ ((𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆) ↔ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))))
153	136, 152	mpbi 229	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
154	153	fveq2d 6894	. . . . . . . 8 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))))
155	153	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
156	155	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
157	156	oveqd 7430	. . . . . . . 8 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))
158	153	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘𝑆) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))))
159	158	fveq1d 6892	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
160	154, 157, 159	oveq123d 7434	. . . . . . 7 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))
161	160	eceq1d 8763	. . . . . 6 ⊢ (𝜑 → [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿))
162	147	eqcomd 2732	. . . . . . 7 ⊢ (𝜑 → (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
163	162	eceq2d 8766	. . . . . 6 ⊢ (𝜑 → [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
164	161, 163	eqtrd 2766	. . . . 5 ⊢ (𝜑 → [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
165	149, 150, 164	3eqtrd 2770	. . . 4 ⊢ (𝜑 → [(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
166	165	fveq2d 6894	. . 3 ⊢ (𝜑 → (𝐼‘[(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = (𝐼‘[((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)))
167		eceq1 8762	. . . . . 6 ⊢ (𝑢 = ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) → [𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
168	167	fveq2d 6894	. . . . 5 ⊢ (𝑢 = ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) → (𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = (𝐼‘[((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)))
169		fveq2 6890	. . . . 5 ⊢ (𝑢 = ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) → ((𝐻 ∘ 𝐹)‘𝑢) = ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
170	168, 169	eqeq12d 2742	. . . 4 ⊢ (𝑢 = ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) → ((𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢) ↔ (𝐼‘[((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
171	56, 57, 131, 27, 42	mulgnn0cld 19082	. . . . 5 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
172	28, 29, 38, 171, 54	grpcld 18934	. . . 4 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
173	170, 129, 172	rspcdva 3608	. . 3 ⊢ (𝜑 → (𝐼‘[((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
174	134, 166, 173	3eqtrd 2770	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
175	75, 172	fvco3d 6991	. . 3 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = (𝐻‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
176		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → 𝑟 = (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
177	176	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → ((eval₁‘𝐾)‘𝑟) = ((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
178	177	fveq1d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) → (((eval₁‘𝐾)‘𝑟)‘𝑀) = (((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀))
179	75, 172	ffvelcdmd 7088	. . . . 5 ⊢ (𝜑 → (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) ∈ (Base‘(Poly₁‘𝐾)))
180		fvexd 6905	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) ∈ V)
181	77, 178, 179, 180	fvmptd 7005	. . . 4 ⊢ (𝜑 → (𝐻‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀))
182	28, 29, 86	ghmlin 19208	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)) ∧ (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∧ ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
183	85, 171, 54, 182	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))
184	183	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → ((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = ((eval₁‘𝐾)‘((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))))
185	184	fveq1d 6892	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) = (((eval₁‘𝐾)‘((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀))
186		eqid 2726	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
187	23, 186	rhmmhm 20454	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘(Poly₁‘𝐾))))
188	72, 187	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘(Poly₁‘𝐾))))
189		eqid 2726	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
190	56, 57, 189	mhmmulg 19102	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘(Poly₁‘𝐾))) ∧ 𝑁 ∈ ℕ₀ ∧ (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))) → (𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))))
191	188, 27, 42, 190	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))))
192	191, 91	oveq12d 7431	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = ((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))))
193	192	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → ((eval₁‘𝐾)‘((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = ((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)))))
194	193	fveq1d 6892	. . . . . 6 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) = (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))))‘𝑀))
195	90	oveq2d 7429	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
196	195, 93	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))) = ((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))
197	196	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → ((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)))) = ((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴))))
198	197	fveq1d 6892	. . . . . . . 8 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))))‘𝑀) = (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))‘𝑀))
199		eqid 2726	. . . . . . . . . . 11 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
200	199, 89, 18, 60, 73, 8, 21	evl1vard 22322	. . . . . . . . . . . 12 ⊢ (𝜑 → ((var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(var₁‘𝐾))‘𝑀) = 𝑀))
201	199, 60, 18, 73, 8, 21, 200, 189, 14, 27	evl1expd 22330	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))‘𝑀) = (𝑁(._g‘(mulGrp‘𝐾))𝑀)))
202	60	ply1crng 22181	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
203	8, 202	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
204	203	crngringd 20222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
205	92	zrhrhm 21494	. . . . . . . . . . . . . . 15 ⊢ ((Poly₁‘𝐾) ∈ Ring → (ℤRHom‘(Poly₁‘𝐾)) ∈ (ℤ_ring RingHom (Poly₁‘𝐾)))
206	49, 73	rhmf 20460	. . . . . . . . . . . . . . 15 ⊢ ((ℤRHom‘(Poly₁‘𝐾)) ∈ (ℤ_ring RingHom (Poly₁‘𝐾)) → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
207	205, 206	syl 17	. . . . . . . . . . . . . 14 ⊢ ((Poly₁‘𝐾) ∈ Ring → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
208	204, 207	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
209	208, 46	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘𝐾))‘𝐴) ∈ (Base‘(Poly₁‘𝐾)))
210		eqidd 2727	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
211	209, 210	jca 510	. . . . . . . . . . 11 ⊢ (𝜑 → (((ℤRHom‘(Poly₁‘𝐾))‘𝐴) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀)))
212		eqid 2726	. . . . . . . . . . 11 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
213	199, 60, 18, 73, 8, 21, 201, 211, 86, 212	evl1addd 22326	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))‘𝑀) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)(((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))))
214	213	simprd 494	. . . . . . . . 9 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))‘𝑀) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)(((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀)))
215	96	zrhrhm 21494	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
216	49, 18	rhmf 20460	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
217	215, 216	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
218	61, 217	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
219	218, 46	ffvelcdmd 7088	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
220	199, 60, 18, 95, 73, 8, 219, 21	evl1scad 22320	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴)))
221	220	simprd 494	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
222	221	eqcomd 2732	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀))
223	97	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) = ((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))
224	223	fveq1d 6892	. . . . . . . . . . 11 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
225	222, 224	eqtr2d 2767	. . . . . . . . . 10 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
226	225	oveq2d 7429	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)(((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀)) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
227	214, 226	eqtrd 2766	. . . . . . . 8 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))(+_g‘(Poly₁‘𝐾))((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))‘𝑀) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
228	198, 227	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))))‘𝑀) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
229	11	cmnmndd 19795	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
230	19, 14, 229, 27, 21	mulgnn0cld 19082	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾))
231	199, 89, 18, 60, 73, 8, 230	evl1vard 22322	. . . . . . . . 9 ⊢ (𝜑 → ((var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(var₁‘𝐾))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)) = (𝑁(._g‘(mulGrp‘𝐾))𝑀)))
232	199, 60, 18, 95, 73, 8, 219, 230	evl1scad 22320	. . . . . . . . 9 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)) = ((ℤRHom‘𝐾)‘𝐴)))
233	199, 60, 18, 73, 8, 230, 231, 232, 86, 212	evl1addd 22326	. . . . . . . 8 ⊢ (𝜑 → (((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴))))
234	233	simprd 494	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)) = ((𝑁(._g‘(mulGrp‘𝐾))𝑀)(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐴)))
235	228, 234	eqtr4d 2769	. . . . . 6 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))))‘𝑀) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))
236	194, 235	eqtrd 2766	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝐾)‘((𝐹‘(𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))(+_g‘(Poly₁‘𝐾))(𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))
237	185, 236	eqtrd 2766	. . . 4 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))))‘𝑀) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))
238	181, 237	eqtrd 2766	. . 3 ⊢ (𝜑 → (𝐻‘(𝐹‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))
239	175, 238	eqtrd 2766	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹)‘((𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))
240	106, 174, 239	3eqtrd 2770	1 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(._g‘(mulGrp‘𝐾))𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 {csn 4623 ∪ cuni 4905 class class class wbr 5143 {copab 5205 ↦ cmpt 5226 “ cima 5675 ∘ ccom 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 [cec 8721 ℕcn 12255 ℕ₀cn0 12515 ℤcz 12601 ∥ cdvds 16248 ℙcprime 16664 Basecbs 17205 +_gcplusg 17258 0_gc0g 17446 /_s cqus 17512 Mndcmnd 18719 MndHom cmhm 18763 Grpcgrp 18920 -_gcsg 18922 ._gcmg 19054 ~_QG cqg 19109 GrpHom cghm 19199 CMndccmn 19771 mulGrpcmgp 20110 1_rcur 20157 Ringcrg 20209 CRingccrg 20210 RingHom crh 20444 Fieldcfield 20701 RSpancrsp 21189 ℤ_ringczring 21429 ℤRHomczrh 21482 chrcchr 21484 ℤ/nℤczn 21485 algSccascl 21843 var₁cv1 22158 Poly₁cpl1 22159 eval₁ce1 22299 PrimRoots cprimroots 41800

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 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 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7992 df-2nd 7993 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-ec 8725 df-qs 8729 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9396 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-z 12602 df-dec 12721 df-uz 12866 df-rp 13020 df-fz 13530 df-fzo 13673 df-fl 13803 df-mod 13881 df-seq 14013 df-exp 14073 df-hash 14340 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-dvds 16249 df-prm 16665 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-0g 17448 df-gsum 17449 df-prds 17454 df-pws 17456 df-imas 17515 df-qus 17516 df-mre 17591 df-mrc 17592 df-acs 17594 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-mhm 18765 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-mulg 19055 df-subg 19110 df-nsg 19111 df-eqg 19112 df-ghm 19200 df-cntz 19304 df-od 19519 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-srg 20163 df-ring 20211 df-cring 20212 df-oppr 20309 df-dvdsr 20332 df-rhm 20447 df-subrng 20521 df-subrg 20546 df-field 20703 df-lmod 20831 df-lss 20902 df-lsp 20942 df-sra 21144 df-rgmod 21145 df-lidl 21190 df-rsp 21191 df-2idl 21232 df-cnfld 21337 df-zring 21430 df-zrh 21486 df-chr 21488 df-zn 21489 df-assa 21844 df-asp 21845 df-ascl 21846 df-psr 21899 df-mvr 21900 df-mpl 21901 df-opsr 21903 df-evls 22080 df-evl 22081 df-psr1 22162 df-vr1 22163 df-ply1 22164 df-coe1 22165 df-evls1 22300 df-evl1 22301 df-primroots 41801

This theorem is referenced by: aks5lem4a 41899

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