aks5lem3a - Mathbox for metakunt

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Theorem aks5lem3a 42146

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 20764	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
9		eqid 2740	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
10	9	crngmgp 20268	. . . . . . . . . . 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 12613	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
14		eqid 2740	. . . . . . . . . 10 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
15	11, 13, 14	isprimroot 42050	. . . . . . . . 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 232	. . . . . . . 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 1142	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
18		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	9, 18	mgpbas 20167	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
20	19	eqcomi 2749	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘𝐾)
21	17, 20	eleqtrdi 2854	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
22	1, 2, 3, 4, 5, 6, 21	aks5lem1 42143	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
23		eqid 2740	. . . . . 6 ⊢ (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
24	23, 9	rhmmhm 20505	. . . . 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 1143	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12613	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		eqid 2740	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))
29		eqid 2740	. . . . 5 ⊢ (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
30		eqid 2740	. . . . . . . . . 10 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
31	30	zncrng 21586	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ CRing)
33		eqid 2740	. . . . . . . . 9 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = (Poly₁‘(ℤ/nℤ‘𝑁))
34	33	ply1crng 22221	. . . . . . . 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 20273	. . . . . 6 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring)
37		ringgrp 20265	. . . . . 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 20273	. . . . . 6 ⊢ (𝜑 → (ℤ/nℤ‘𝑁) ∈ Ring)
40		eqid 2740	. . . . . . 7 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁))
41	40, 33, 28	vr1cl 22240	. . . . . 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 2740	. . . . . . 7 ⊢ (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))
44		eqid 2740	. . . . . . 7 ⊢ (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))
45		eqid 2740	. . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁))
46		aks5lem3a.12	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
47	33, 43, 44, 45, 32, 46	ply1asclzrhval 42145	. . . . . 6 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴))
48	44	zrhrhm 21545	. . . . . . . . 9 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) ∈ Ring → (ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁))) ∈ (ℤ_ring RingHom (Poly₁‘(ℤ/nℤ‘𝑁))))
49		zringbas 21487	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
50	49, 28	rhmf 20511	. . . . . . . . 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 7119	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
54	47, 53	eqeltrd 2844	. . . . 5 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))))
55	28, 29, 38, 42, 54	grpcld 18987	. . . 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 20167	. . . . 5 ⊢ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (Base‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
57		eqid 2740	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
58	56, 57, 14	mhmmulg 19155	. . . 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 1371	. . 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 2740	. . . . . . . 8 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
61	8	crngringd 20273	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
62	2	eqcomi 2749	. . . . . . . . . 10 ⊢ (chr‘𝐾) = 𝑃
63	3	simp1d 1142	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℙ)
64		prmnn 16721	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
65	63, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
66	65	nnzd 12666	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	62, 66	eqeltrid 2848	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∈ ℤ)
68	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (chr‘𝐾) = 𝑃)
69	3	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
70	68, 69	eqbrtrd 5188	. . . . . . . . 9 ⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁)
71	61, 26, 67, 70, 30, 5	zndvdchrrhm 41927	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((ℤ/nℤ‘𝑁) RingHom 𝐾))
72	33, 60, 28, 4, 71	rhmply1 22411	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((Poly₁‘(ℤ/nℤ‘𝑁)) RingHom (Poly₁‘𝐾)))
73		eqid 2740	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
74	28, 73	rhmf 20511	. . . . . . 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 7022	. . . . 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 484	. . . . . . . . 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 6924	. . . . . . . 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 6922	. . . . . . 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 7119	. . . . . . 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 6935	. . . . . . 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 7036	. . . . . 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 20510	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
87	28, 29, 86	ghmlin 19261	. . . . . . . . . 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 1371	. . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
90	33, 60, 28, 4, 40, 89, 71	rhmply1vr1 22412	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(var₁‘(ℤ/nℤ‘𝑁))) = (var₁‘𝐾))
91	47	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)))
92		eqid 2740	. . . . . . . . . . . . 13 ⊢ (ℤRHom‘(Poly₁‘𝐾)) = (ℤRHom‘(Poly₁‘𝐾))
93	72, 46, 44, 92	rhmzrhval 41926	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘((ℤRHom‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
94	91, 93	eqtrd 2780	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
95		eqid 2740	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
96		eqid 2740	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
97	60, 95, 92, 96, 8, 46	ply1asclzrhval 42145	. . . . . . . . . . 11 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘(Poly₁‘𝐾))‘𝐴))
98	94, 97	eqtr4d 2783	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))
99	90, 98	oveq12d 7466	. . . . . . . . 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 2780	. . . . . . . 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 6924	. . . . . . 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 6922	. . . . . 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 2780	. . . . 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 2780	. . . 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 7464	. . 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 2781	. 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 8802	. . . . . . 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 6924	. . . . . 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 6920	. . . . . 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 2756	. . . . 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 7458	. . . . . . . . 9 ⊢ (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)
115	113, 114	oveq12i 7460	. . . . . . . 8 ⊢ (𝑆 /_s (𝑆 ~_QG 𝐿)) = ((Poly₁‘(ℤ/nℤ‘𝑁)) /_s ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
116	112, 115	eqtri 2768	. . . . . . 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 6923	. . . . . . . . 9 ⊢ (RSpan‘𝑆) = (RSpan‘(Poly₁‘(ℤ/nℤ‘𝑁)))
119	113	fveq2i 6923	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))
120	119	fveq2i 6923	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
121	120	oveqi 7461	. . . . . . . . . . 11 ⊢ (𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑅(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁)))
122	113	fveq2i 6923	. . . . . . . . . . 11 ⊢ (1_r‘𝑆) = (1_r‘(Poly₁‘(ℤ/nℤ‘𝑁)))
123	113	fveq2i 6923	. . . . . . . . . . 11 ⊢ (-_g‘𝑆) = (-_g‘(Poly₁‘(ℤ/nℤ‘𝑁)))
124	121, 122, 123	oveq123i 7462	. . . . . . . . . 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 4659	. . . . . . . . 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 6926	. . . . . . . 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 2768	. . . . . . 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 42144	. . . . . 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 495	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘(Poly₁‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑢]((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑢))
130	23	ringmgp 20266	. . . . . . 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 19135	. . . . 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 3636	. . . 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 2746	. . 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 2749	. . . . . . . . . . 11 ⊢ (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆
136	135	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆)
137	136	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (mulGrp‘𝑆))
138	137	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))) = (._g‘(mulGrp‘𝑆)))
139		eqidd 2741	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = 𝑁)
140	136	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (+_g‘𝑆))
141		eqidd 2741	. . . . . . . . 9 ⊢ (𝜑 → (var₁‘(ℤ/nℤ‘𝑁)) = (var₁‘(ℤ/nℤ‘𝑁)))
142	136	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))) = (algSc‘𝑆))
143	142	fveq1d 6922	. . . . . . . . 9 ⊢ (𝜑 → ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
144	140, 141, 143	oveq123d 7469	. . . . . . . 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 7469	. . . . . . 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 8803	. . . . . 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 7463	. . . . . . 7 ⊢ (𝜑 → ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿) = (𝑆 ~_QG 𝐿))
148	147	eceq2d 8806	. . . . . 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 2780	. . . . 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 2747	. . . . . . . . . . 11 ⊢ ((Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆 ↔ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
152	151	imbi2i 336	. . . . . . . . . 10 ⊢ ((𝜑 → (Poly₁‘(ℤ/nℤ‘𝑁)) = 𝑆) ↔ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))))
153	136, 152	mpbi 230	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁)))
154	153	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘(Poly₁‘(ℤ/nℤ‘𝑁))))
155	153	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑆) = (mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))
156	155	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁)))))
157	156	oveqd 7465	. . . . . . . 8 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))) = (𝑁(._g‘(mulGrp‘(Poly₁‘(ℤ/nℤ‘𝑁))))(var₁‘(ℤ/nℤ‘𝑁))))
158	153	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘𝑆) = (algSc‘(Poly₁‘(ℤ/nℤ‘𝑁))))
159	158	fveq1d 6922	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘(Poly₁‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))
160	154, 157, 159	oveq123d 7469	. . . . . . 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 8803	. . . . . 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 2746	. . . . . . 7 ⊢ (𝜑 → (𝑆 ~_QG 𝐿) = ((Poly₁‘(ℤ/nℤ‘𝑁)) ~_QG 𝐿))
163	162	eceq2d 8806	. . . . . 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 2780	. . . . 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 2784	. . . 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 6924	. . 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 8802	. . . . . 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 6924	. . . . 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 6920	. . . . 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 2756	. . . 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 19135	. . . . 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 18987	. . . 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 3636	. . 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 2784	. 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 7022	. . 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 484	. . . . . . 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 6924	. . . . . 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 6922	. . . . 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 7119	. . . . 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 6935	. . . . 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 7036	. . . 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 19261	. . . . . . . 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 1371	. . . . . . 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 6924	. . . . . 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 6922	. . . . 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 2740	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
187	23, 186	rhmmhm 20505	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
190	56, 57, 189	mhmmulg 19155	. . . . . . . . . 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 1371	. . . . . . . . 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 7466	. . . . . . . 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 6924	. . . . . . 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 6922	. . . . . 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 7464	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(𝐹‘(var₁‘(ℤ/nℤ‘𝑁)))) = (𝑁(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
196	195, 93	oveq12d 7466	. . . . . . . . . 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 6924	. . . . . . . . 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 6922	. . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
200	199, 89, 18, 60, 73, 8, 21	evl1vard 22362	. . . . . . . . . . . 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 22370	. . . . . . . . . . 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 22221	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
203	8, 202	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
204	203	crngringd 20273	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
205	92	zrhrhm 21545	. . . . . . . . . . . . . . 15 ⊢ ((Poly₁‘𝐾) ∈ Ring → (ℤRHom‘(Poly₁‘𝐾)) ∈ (ℤ_ring RingHom (Poly₁‘𝐾)))
206	49, 73	rhmf 20511	. . . . . . . . . . . . . . 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 7119	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℤRHom‘(Poly₁‘𝐾))‘𝐴) ∈ (Base‘(Poly₁‘𝐾)))
210		eqidd 2741	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
211	209, 210	jca 511	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
213	199, 60, 18, 73, 8, 21, 201, 211, 86, 212	evl1addd 22366	. . . . . . . . . 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 495	. . . . . . . . 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 21545	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
216	49, 18	rhmf 20511	. . . . . . . . . . . . . . . . 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 7119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
220	199, 60, 18, 95, 73, 8, 219, 21	evl1scad 22360	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
222	221	eqcomd 2746	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀))
223	97	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) = ((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴)))
224	223	fveq1d 6922	. . . . . . . . . . 11 ⊢ (𝜑 → (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀))
225	222, 224	eqtr2d 2781	. . . . . . . . . 10 ⊢ (𝜑 → (((eval₁‘𝐾)‘((ℤRHom‘(Poly₁‘𝐾))‘𝐴))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴))
226	225	oveq2d 7464	. . . . . . . . 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 2780	. . . . . . . 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 2780	. . . . . . 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 19846	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
230	19, 14, 229, 27, 21	mulgnn0cld 19135	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(._g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾))
231	199, 89, 18, 60, 73, 8, 230	evl1vard 22362	. . . . . . . . 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 22360	. . . . . . . . 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 22366	. . . . . . . 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 495	. . . . . . 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 2783	. . . . . 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 2780	. . . . 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 2780	. . . 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 2780	. . 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 2780	. 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 2784	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 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 {csn 4648 ∪ cuni 4931 class class class wbr 5166 {copab 5228 ↦ cmpt 5249 “ cima 5703 ∘ ccom 5704 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 [cec 8761 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ∥ cdvds 16302 ℙcprime 16718 Basecbs 17258 +_gcplusg 17311 0_gc0g 17499 /_s cqus 17565 Mndcmnd 18772 MndHom cmhm 18816 Grpcgrp 18973 -_gcsg 18975 ._gcmg 19107 ~_QG cqg 19162 GrpHom cghm 19252 CMndccmn 19822 mulGrpcmgp 20161 1_rcur 20208 Ringcrg 20260 CRingccrg 20261 RingHom crh 20495 Fieldcfield 20752 RSpancrsp 21240 ℤ_ringczring 21480 ℤRHomczrh 21533 chrcchr 21535 ℤ/nℤczn 21536 algSccascl 21895 var₁cv1 22198 Poly₁cpl1 22199 eval₁ce1 22339 PrimRoots cprimroots 42048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-prm 16719 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-imas 17568 df-qus 17569 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-field 20754 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-chr 21539 df-zn 21540 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-evls1 22340 df-evl1 22341 df-primroots 42049

This theorem is referenced by: aks5lem4a 42147

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