aks5lem3a - Mathbox for metakunt

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Theorem aks5lem3a 42956

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 20842	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
9		eqid 2763	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
10	9	crngmgp 20318	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
11	8, 10	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
12		aks5lema.11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	nnnn0d 12560	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		eqid 2763	. . . . . . . . . 10 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
15	11, 13, 14	isprimroot 42860	. . . . . . . . 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 235	. . . . . . . 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 1160	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
18		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	9, 18	mgpbas 20216	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
20	19	eqcomi 2772	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝐾)
21	17, 20	eleqtrdi 2873	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
22	1, 2, 3, 4, 5, 6, 21	aks5lem1 42953	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
23		eqid 2763	. . . . . 6 ⊢ (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
24	23, 9	rhmmhm 20558	. . . . 5 ⊢ ((𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾) → (𝐻 ∘ 𝐹) ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘𝐾)))
25	22, 24	syl 18	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘𝐾)))
26	3	simp2d 1161	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12560	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		eqid 2763	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))
29		eqid 2763	. . . . 5 ⊢ (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
30		eqid 2763	. . . . . . . . . 10 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
31	30	zncrng 21694	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
32	27, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ CRing)
33		eqid 2763	. . . . . . . . 9 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = (Poly₁‘(ℤ/nℤ‘𝑁))
34	33	ply1crng 22358	. . . . . . . 8 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
35	32, 34	syl 18	. . . . . . 7 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ CRing)
36	35	crngringd 20323	. . . . . 6 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring)
37		ringgrp 20315	. . . . . 6 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
38	36, 37	syl 18	. . . . 5 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Grp)
39	32	crngringd 20323	. . . . . 6 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ Ring)
40		eqid 2763	. . . . . . 7 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁))
41	40, 33, 28	vr1cl 22377	. . . . . 6 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
42	39, 41	syl 18	. . . . 5 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
43		eqid 2763	. . . . . . 7 ⊢ (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))
44		eqid 2763	. . . . . . 7 ⊢ (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))
45		eqid 2763	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
46		aks5lem3a.12	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
47	33, 43, 44, 45, 32, 46	ply1asclzrhval 42955	. . . . . 6 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))
48	44	zrhrhm 21661	. . . . . . . . 9 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (ℤ_ring RingHom (Poly₁‘(ℤ/nℤ‘𝑁))))
49		zringbas 21603	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
50	49, 28	rhmf 20563	. . . . . . . . 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 18	. . . . . . . 8 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))):ℤ⟶(Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
52	36, 51	syl 18	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))):ℤ⟶(Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
53	52, 46	ffvelcdmd 7080	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
54	47, 53	eqeltrd 2863	. . . . 5 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
55	28, 29, 38, 42, 54	grpcld 19009	. . . 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 20216	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
57		eqid 2763	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
58	56, 57, 14	mhmmulg 19176	. . . 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 1398	. . 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 2763	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
61	8	crngringd 20323	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
62	2	eqcomi 2772	. . . . . . . . . 10 ⊢ (chr‘𝐾) = 𝑃
63	3	simp1d 1160	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
64		prmnn 16727	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
65	63, 64	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
66	65	nnzd 12612	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	62, 66	eqeltrid 2867	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∈ ℤ)
68	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (chr‘𝐾) = 𝑃)
69	3	simp3d 1162	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
70	68, 69	eqbrtrd 5133	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁)
71	61, 26, 67, 70, 30, 5	zndvdchrrhm 42740	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾))
72	33, 60, 28, 4, 71	rhmply1 22543	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)))
73		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
74	28, 73	rhmf 20563	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
75	72, 74	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))⟶(Base‘(Poly₁‘𝐾)))
76	75, 55	fvco3d 6982	. . . . 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 489	. . . . . . . . 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 6885	. . . . . . . 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 6883	. . . . . . 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 7080	. . . . . . 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 6896	. . . . . . 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 6997	. . . . . 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 20562	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
85	72, 84	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) GrpHom (Poly₁‘𝐾)))
86		eqid 2763	. . . . . . . . . . 11 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
87	28, 29, 86	ghmlin 19286	. . . . . . . . . 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 1398	. . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
90	33, 60, 28, 4, 40, 89, 71	rhmply1vr1 22544	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(var₁‘(ℤ/nℤ‘𝑁))) = (var₁‘𝐾))
91	47	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)))
92		eqid 2763	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘(Poly₁‘𝐾)) = (ℤRHom‘(Poly₁‘𝐾))
93	72, 46, 44, 92	rhmzrhval 42739	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
94	91, 93	eqtrd 2798	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
95		eqid 2763	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
96		eqid 2763	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
97	60, 95, 92, 96, 8, 46	ply1asclzrhval 42955	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
98	94, 97	eqtr4d 2801	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))
99	90, 98	oveq12d 7428	. . . . . . . . 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 2798	. . . . . . . 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 6885	. . . . . . 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 6883	. . . . . 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 2798	. . . . 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 2798	. . . 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 7426	. . 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 2799	. 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 8730	. . . . . . 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 6885	. . . . . 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 6881	. . . . . 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 2779	. . . . 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 7420	. . . . . . . . 9 ⊢ (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)
115	113, 114	oveq12i 7422	. . . . . . . 8 ⊢ (𝑆 /_s (𝑆 ~_QG 𝐿)) = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
116	112, 115	eqtri 2786	. . . . . . 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 6884	. . . . . . . . 9 ⊢ (RSpan‘𝑆) = (RSpan‘(Poly₁‘(ℤ/nℤ‘𝑁)))
119	113	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
120	119	fveq2i 6884	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
121	120	oveqi 7423	. . . . . . . . . . 11 ⊢ (𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))
122	113	fveq2i 6884	. . . . . . . . . . 11 ⊢ (1_r‘𝑆) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
123	113	fveq2i 6884	. . . . . . . . . . 11 ⊢ (-_g‘𝑆) = (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
124	121, 122, 123	oveq123i 7424	. . . . . . . . . 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 4600	. . . . . . . . 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 6887	. . . . . . . 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 2786	. . . . . . 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 42954	. . . . . 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 500	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢))
130	23	ringmgp 20316	. . . . . . 7 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
131	36, 130	syl 18	. . . . . 6 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ Mnd)
132	56, 57, 131, 27, 55	mulgnn0cld 19156	. . . . 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 3582	. . . 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 2769	. . 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 2772	. . . . . . . . . . 11 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆
136	135	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆)
137	136	fveq2d 6885	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘𝑆))
138	137	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘𝑆)))
139		eqidd 2764	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = 𝑁)
140	136	fveq2d 6885	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘𝑆))
141		eqidd 2764	. . . . . . . . 9 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁)))
142	136	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘𝑆))
143	142	fveq1d 6883	. . . . . . . . 9 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
144	140, 141, 143	oveq123d 7431	. . . . . . . 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 7431	. . . . . . 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 8731	. . . . . 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 7425	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = (𝑆 ~_QG 𝐿))
148	147	eceq2d 8734	. . . . . 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 2798	. . . . 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 2770	. . . . . . . . . . 11 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆 ↔ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
152	151	imbi2i 339	. . . . . . . . . 10 ⊢ ((𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆) ↔ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))))
153	136, 152	mpbi 233	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
154	153	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))))
155	153	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
156	155	fveq2d 6885	. . . . . . . . 9 ⊢ (𝜑 → (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
157	156	oveqd 7427	. . . . . . . 8 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))
158	153	fveq2d 6885	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘𝑆) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))))
159	158	fveq1d 6883	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
160	154, 157, 159	oveq123d 7431	. . . . . . 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 8731	. . . . . 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 2769	. . . . . . 7 ⊢ (𝜑 → (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
163	162	eceq2d 8734	. . . . . 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 2798	. . . . 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 2802	. . . 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 6885	. . 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 8730	. . . . . 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 6885	. . . . 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 6881	. . . . 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 2779	. . . 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 19156	. . . . 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 19009	. . . 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 3582	. . 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 2802	. 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 6982	. . 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 489	. . . . . . 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 6885	. . . . . 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 6883	. . . . 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 7080	. . . . 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 6896	. . . . 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 6997	. . . 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 19286	. . . . . . . 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 1398	. . . . . . 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 6885	. . . . . 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 6883	. . . . 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 2763	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
187	23, 186	rhmmhm 20558	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)) → 𝐹 ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘(Poly₁‘𝐾))))
188	72, 187	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) MndHom (mulGrp‘(Poly₁‘𝐾))))
189		eqid 2763	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
190	56, 57, 189	mhmmulg 19176	. . . . . . . . . 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 1398	. . . . . . . . 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 7428	. . . . . . . 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 6885	. . . . . . 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 6883	. . . . . 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 7426	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
196	195, 93	oveq12d 7428	. . . . . . . . . 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 6885	. . . . . . . . 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 6883	. . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
200	199, 89, 18, 60, 73, 8, 21	evl1vard 22497	. . . . . . . . . . . 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 22505	. . . . . . . . . . 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 22358	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
203	8, 202	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
204	203	crngringd 20323	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
205	92	zrhrhm 21661	. . . . . . . . . . . . . . 15 ⊢ ((Poly₁‘𝐾) ∈ Ring → (ℤRHom‘(Poly₁‘𝐾)) ∈ (ℤ_ring RingHom (Poly₁‘𝐾)))
206	49, 73	rhmf 20563	. . . . . . . . . . . . . . 15 ⊢ ((ℤRHom‘(Poly₁‘𝐾)) ∈ (ℤ_ring RingHom (Poly₁‘𝐾)) → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
207	205, 206	syl 18	. . . . . . . . . . . . . 14 ⊢ ((Poly₁‘𝐾) ∈ Ring → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
208	204, 207	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤRHom‘(Poly₁‘𝐾)):ℤ⟶(Base‘(Poly₁‘𝐾)))
209	208, 46	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘𝐾))‘𝐴) ∈ (Base‘(Poly₁‘𝐾)))
210		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
211	209, 210	jca 520	. . . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
213	199, 60, 18, 73, 8, 21, 201, 211, 86, 212	evl1addd 22501	. . . . . . . . . 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 500	. . . . . . . . 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 21661	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
216	49, 18	rhmf 20563	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
217	215, 216	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
218	61, 217	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
219	218, 46	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
220	199, 60, 18, 95, 73, 8, 219, 21	evl1scad 22495	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
222	221	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀))
223	97	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) = ((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))
224	223	fveq1d 6883	. . . . . . . . . . 11 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
225	222, 224	eqtr2d 2799	. . . . . . . . . 10 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
226	225	oveq2d 7426	. . . . . . . . 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 2798	. . . . . . . 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 2798	. . . . . . 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 19869	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
230	19, 14, 229, 27, 21	mulgnn0cld 19156	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾))
231	199, 89, 18, 60, 73, 8, 230	evl1vard 22497	. . . . . . . . 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 22495	. . . . . . . . 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 22501	. . . . . . . 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 500	. . . . . . 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 2801	. . . . . 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 2798	. . . . 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 2798	. . . 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 2798	. . 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 2798	. 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 2802	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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 {csn 4589 ∪ cuni 4872 class class class wbr 5109 {copab 5173 ↦ cmpt 5192 “ cima 5664 ∘ ccom 5665 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 [cec 8688 ℕcn 12228 ℕ₀cn0 12499 ℤcz 12586 ∥ cdvds 16305 ℙcprime 16724 Basecbs 17264 +_gcplusg 17305 0_gc0g 17487 /_s cqus 17554 Mndcmnd 18787 MndHom cmhm 18834 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 RSpancrsp 21331 ℤ_ringczring 21596 ℤRHomczrh 21649 chrcchr 21651 ℤ/nℤczn 21652 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: aks5lem4a 42957

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