| 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 20743 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ CRing) | 
| 9 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(mulGrp‘𝐾) =
(mulGrp‘𝐾) | 
| 10 | 9 | crngmgp 20239 | . . . . . . . . . . 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 12589 | . . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈
ℕ0) | 
| 14 |  | eqid 2736 | . . . . . . . . . 10
⊢
(.g‘(mulGrp‘𝐾)) =
(.g‘(mulGrp‘𝐾)) | 
| 15 | 11, 13, 14 | isprimroot 42095 | . . . . . . . . 9
⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑑 ∈ ℕ0
((𝑑(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑑)))) | 
| 16 | 7, 15 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑑 ∈ ℕ0
((𝑑(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑑))) | 
| 17 | 16 | simp1d 1142 | . . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾))) | 
| 18 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) | 
| 19 | 9, 18 | mgpbas 20143 | . . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘(mulGrp‘𝐾)) | 
| 20 | 19 | eqcomi 2745 | . . . . . . 7
⊢
(Base‘(mulGrp‘𝐾)) = (Base‘𝐾) | 
| 21 | 17, 20 | eleqtrdi 2850 | . . . . . 6
⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾)) | 
| 22 | 1, 2, 3, 4, 5, 6, 21 | aks5lem1 42188 | . . . . 5
⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈
((Poly1‘(ℤ/nℤ‘𝑁)) RingHom 𝐾)) | 
| 23 |  | eqid 2736 | . . . . . 6
⊢
(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 24 | 23, 9 | rhmmhm 20480 | . . . . 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 12589 | . . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 28 |  | eqid 2736 | . . . . 5
⊢
(Base‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(Base‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 29 |  | eqid 2736 | . . . . 5
⊢
(+g‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(+g‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 30 |  | eqid 2736 | . . . . . . . . . 10
⊢
(ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | 
| 31 | 30 | zncrng 21564 | . . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (ℤ/nℤ‘𝑁) ∈ CRing) | 
| 32 | 27, 31 | syl 17 | . . . . . . . 8
⊢ (𝜑 →
(ℤ/nℤ‘𝑁) ∈ CRing) | 
| 33 |  | eqid 2736 | . . . . . . . . 9
⊢
(Poly1‘(ℤ/nℤ‘𝑁)) =
(Poly1‘(ℤ/nℤ‘𝑁)) | 
| 34 | 33 | ply1crng 22201 | . . . . . . . 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 20244 | . . . . . 6
⊢ (𝜑 →
(Poly1‘(ℤ/nℤ‘𝑁)) ∈ Ring) | 
| 37 |  | ringgrp 20236 | . . . . . 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 20244 | . . . . . 6
⊢ (𝜑 →
(ℤ/nℤ‘𝑁) ∈ Ring) | 
| 40 |  | eqid 2736 | . . . . . . 7
⊢
(var1‘(ℤ/nℤ‘𝑁)) =
(var1‘(ℤ/nℤ‘𝑁)) | 
| 41 | 40, 33, 28 | vr1cl 22220 | . . . . . 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 2736 | . . . . . . 7
⊢
(algSc‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(algSc‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 44 |  | eqid 2736 | . . . . . . 7
⊢
(ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 45 |  | eqid 2736 | . . . . . . 7
⊢
(ℤRHom‘(ℤ/nℤ‘𝑁)) =
(ℤRHom‘(ℤ/nℤ‘𝑁)) | 
| 46 |  | aks5lem3a.12 | . . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℤ) | 
| 47 | 33, 43, 44, 45, 32, 46 | ply1asclzrhval 42190 | . . . . . 6
⊢ (𝜑 →
((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) =
((ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁)))‘𝐴)) | 
| 48 | 44 | zrhrhm 21523 | . . . . . . . . 9
⊢
((Poly1‘(ℤ/nℤ‘𝑁)) ∈ Ring →
(ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁))) ∈
(ℤring RingHom
(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 49 |  | zringbas 21465 | . . . . . . . . . 10
⊢ ℤ =
(Base‘ℤring) | 
| 50 | 49, 28 | rhmf 20486 | . . . . . . . . 9
⊢
((ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁))) ∈ (ℤring
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 7104 | . . . . . 6
⊢ (𝜑 →
((ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁)))‘𝐴) ∈
(Base‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 54 | 47, 53 | eqeltrd 2840 | . . . . 5
⊢ (𝜑 →
((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) ∈
(Base‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 55 | 28, 29, 38, 42, 54 | grpcld 18966 | . . . 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 20143 | . . . . 5
⊢
(Base‘(Poly1‘(ℤ/nℤ‘𝑁))) =
(Base‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 57 |  | eqid 2736 | . . . . 5
⊢
(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) =
(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 58 | 56, 57, 14 | mhmmulg 19134 | . . . 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 1372 | . . 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 2736 | . . . . . . . 8
⊢
(Poly1‘𝐾) = (Poly1‘𝐾) | 
| 61 | 8 | crngringd 20244 | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Ring) | 
| 62 | 2 | eqcomi 2745 | . . . . . . . . . 10
⊢
(chr‘𝐾) =
𝑃 | 
| 63 | 3 | simp1d 1142 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| 64 |  | prmnn 16712 | . . . . . . . . . . . 12
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) | 
| 65 | 63, 64 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ ℕ) | 
| 66 | 65 | nnzd 12642 | . . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ ℤ) | 
| 67 | 62, 66 | eqeltrid 2844 | . . . . . . . . 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 5164 | . . . . . . . . 9
⊢ (𝜑 → (chr‘𝐾) ∥ 𝑁) | 
| 71 | 61, 26, 67, 70, 30, 5 | zndvdchrrhm 41973 | . . . . . . . 8
⊢ (𝜑 → 𝐺 ∈
((ℤ/nℤ‘𝑁) RingHom 𝐾)) | 
| 72 | 33, 60, 28, 4, 71 | rhmply1 22391 | . . . . . . 7
⊢ (𝜑 → 𝐹 ∈
((Poly1‘(ℤ/nℤ‘𝑁)) RingHom (Poly1‘𝐾))) | 
| 73 |  | eqid 2736 | . . . . . . . 8
⊢
(Base‘(Poly1‘𝐾)) =
(Base‘(Poly1‘𝐾)) | 
| 74 | 28, 73 | rhmf 20486 | . . . . . . 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 7008 | . . . . 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 6909 | . . . . . . . 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 6907 | . . . . . . 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 7104 | . . . . . . 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 6920 | . . . . . . 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 7022 | . . . . . 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 20485 | . . . . . . . . . . 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 2736 | . . . . . . . . . . 11
⊢
(+g‘(Poly1‘𝐾)) =
(+g‘(Poly1‘𝐾)) | 
| 87 | 28, 29, 86 | ghmlin 19240 | . . . . . . . . . 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 1372 | . . . . . . . . 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 2736 | . . . . . . . . . . 11
⊢
(var1‘𝐾) = (var1‘𝐾) | 
| 90 | 33, 60, 28, 4, 40, 89, 71 | rhmply1vr1 22392 | . . . . . . . . . 10
⊢ (𝜑 → (𝐹‘(var1‘(ℤ/nℤ‘𝑁))) = (var1‘𝐾)) | 
| 91 | 47 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐹‘((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) =
(𝐹‘((ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁)))‘𝐴))) | 
| 92 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(ℤRHom‘(Poly1‘𝐾)) =
(ℤRHom‘(Poly1‘𝐾)) | 
| 93 | 72, 46, 44, 92 | rhmzrhval 41972 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐹‘((ℤRHom‘(Poly1‘(ℤ/nℤ‘𝑁)))‘𝐴)) =
((ℤRHom‘(Poly1‘𝐾))‘𝐴)) | 
| 94 | 91, 93 | eqtrd 2776 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) =
((ℤRHom‘(Poly1‘𝐾))‘𝐴)) | 
| 95 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(algSc‘(Poly1‘𝐾)) =
(algSc‘(Poly1‘𝐾)) | 
| 96 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(ℤRHom‘𝐾) = (ℤRHom‘𝐾) | 
| 97 | 60, 95, 92, 96, 8, 46 | ply1asclzrhval 42190 | . . . . . . . . . . 11
⊢ (𝜑 →
((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)) =
((ℤRHom‘(Poly1‘𝐾))‘𝐴)) | 
| 98 | 94, 97 | eqtr4d 2779 | . . . . . . . . . 10
⊢ (𝜑 → (𝐹‘((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) =
((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) | 
| 99 | 90, 98 | oveq12d 7450 | . . . . . . . . 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 2776 | . . . . . . . 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 6909 | . . . . . . 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 6907 | . . . . . 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 2776 | . . . . 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 2776 | . . . 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 7448 | . . 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 2777 | . 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 8785 | . . . . . . 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 6909 | . . . . . 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 6905 | . . . . . 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 2752 | . . . . 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 7442 | . . . . . . . . 9
⊢ (𝑆 ~QG 𝐿) =
((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿) | 
| 115 | 113, 114 | oveq12i 7444 | . . . . . . . 8
⊢ (𝑆 /s (𝑆 ~QG 𝐿)) =
((Poly1‘(ℤ/nℤ‘𝑁)) /s
((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿)) | 
| 116 | 112, 115 | eqtri 2764 | . . . . . . 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)(1r‘𝑆))}) | 
| 118 | 113 | fveq2i 6908 | . . . . . . . . 9
⊢
(RSpan‘𝑆) =
(RSpan‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 119 | 113 | fveq2i 6908 | . . . . . . . . . . . . 13
⊢
(mulGrp‘𝑆) =
(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 120 | 119 | fveq2i 6908 | . . . . . . . . . . . 12
⊢
(.g‘(mulGrp‘𝑆)) =
(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 121 | 120 | oveqi 7445 | . . . . . . . . . . 11
⊢ (𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁))) = (𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁))) | 
| 122 | 113 | fveq2i 6908 | . . . . . . . . . . 11
⊢
(1r‘𝑆) =
(1r‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 123 | 113 | fveq2i 6908 | . . . . . . . . . . 11
⊢
(-g‘𝑆) =
(-g‘(Poly1‘(ℤ/nℤ‘𝑁))) | 
| 124 | 121, 122,
123 | oveq123i 7446 | . . . . . . . . . 10
⊢ ((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆)) = ((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 125 | 124 | sneqi 4636 | . . . . . . . . 9
⊢ {((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))} = {((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁))))} | 
| 126 | 118, 125 | fveq12i 6911 | . . . . . . . 8
⊢
((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))}) =
((RSpan‘(Poly1‘(ℤ/nℤ‘𝑁)))‘{((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁))))}) | 
| 127 | 117, 126 | eqtri 2764 | . . . . . . 7
⊢ 𝐿 =
((RSpan‘(Poly1‘(ℤ/nℤ‘𝑁)))‘{((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁))))}) | 
| 128 | 1, 2, 3, 4, 5, 6, 7, 111, 116, 127, 12 | aks5lem2 42189 | . . . . . 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 20237 | . . . . . . 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 19114 | . . . . 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 3622 | . . . 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 2742 | . . 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 2745 | . . . . . . . . . . 11
⊢
(Poly1‘(ℤ/nℤ‘𝑁)) = 𝑆 | 
| 136 | 135 | a1i 11 | . . . . . . . . . 10
⊢ (𝜑 →
(Poly1‘(ℤ/nℤ‘𝑁)) = 𝑆) | 
| 137 | 136 | fveq2d 6909 | . . . . . . . . 9
⊢ (𝜑 →
(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))) = (mulGrp‘𝑆)) | 
| 138 | 137 | fveq2d 6909 | . . . . . . . 8
⊢ (𝜑 →
(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) = (.g‘(mulGrp‘𝑆))) | 
| 139 |  | eqidd 2737 | . . . . . . . 8
⊢ (𝜑 → 𝑁 = 𝑁) | 
| 140 | 136 | fveq2d 6909 | . . . . . . . . 9
⊢ (𝜑 →
(+g‘(Poly1‘(ℤ/nℤ‘𝑁))) = (+g‘𝑆)) | 
| 141 |  | eqidd 2737 | . . . . . . . . 9
⊢ (𝜑 →
(var1‘(ℤ/nℤ‘𝑁)) =
(var1‘(ℤ/nℤ‘𝑁))) | 
| 142 | 136 | fveq2d 6909 | . . . . . . . . . 10
⊢ (𝜑 →
(algSc‘(Poly1‘(ℤ/nℤ‘𝑁))) = (algSc‘𝑆)) | 
| 143 | 142 | fveq1d 6907 | . . . . . . . . 9
⊢ (𝜑 →
((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) = ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) | 
| 144 | 140, 141,
143 | oveq123d 7453 | . . . . . . . 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 7453 | . . . . . . 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 8786 | . . . . . 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 7447 | . . . . . . 7
⊢ (𝜑 →
((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿) = (𝑆 ~QG 𝐿)) | 
| 148 | 147 | eceq2d 8789 | . . . . . 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 2776 | . . . . 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 2743 | . . . . . . . . . . 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 6909 | . . . . . . . 8
⊢ (𝜑 → (+g‘𝑆) =
(+g‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 155 | 153 | fveq2d 6909 | . . . . . . . . . 10
⊢ (𝜑 → (mulGrp‘𝑆) =
(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 156 | 155 | fveq2d 6909 | . . . . . . . . 9
⊢ (𝜑 →
(.g‘(mulGrp‘𝑆)) =
(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))) | 
| 157 | 156 | oveqd 7449 | . . . . . . . 8
⊢ (𝜑 → (𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁))) = (𝑁(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))) | 
| 158 | 153 | fveq2d 6909 | . . . . . . . . 9
⊢ (𝜑 → (algSc‘𝑆) =
(algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))) | 
| 159 | 158 | fveq1d 6907 | . . . . . . . 8
⊢ (𝜑 → ((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)) =
((algSc‘(Poly1‘(ℤ/nℤ‘𝑁)))‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) | 
| 160 | 154, 157,
159 | oveq123d 7453 | . . . . . . 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 8786 | . . . . . 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 2742 | . . . . . . 7
⊢ (𝜑 → (𝑆 ~QG 𝐿) =
((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿)) | 
| 163 | 162 | eceq2d 8789 | . . . . . 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 2776 | . . . . 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 2780 | . . . 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 6909 | . . 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 8785 | . . . . . 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 6909 | . . . . 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 6905 | . . . . 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 2752 | . . . 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 19114 | . . . . 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 18966 | . . . 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 3622 | . . 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 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))))) = ((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 7008 | . . 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 6909 | . . . . . 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 6907 | . . . . 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 7104 | . . . . 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 6920 | . . . . 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 7022 | . . . 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 19240 | . . . . . . . 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 1372 | . . . . . . 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 6909 | . . . . . 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 6907 | . . . . 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 2736 | . . . . . . . . . . . 12
⊢
(mulGrp‘(Poly1‘𝐾)) =
(mulGrp‘(Poly1‘𝐾)) | 
| 187 | 23, 186 | rhmmhm 20480 | . . . . . . . . . . 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 2736 | . . . . . . . . . . 11
⊢
(.g‘(mulGrp‘(Poly1‘𝐾))) =
(.g‘(mulGrp‘(Poly1‘𝐾))) | 
| 190 | 56, 57, 189 | mhmmulg 19134 | . . . . . . . . . 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 1372 | . . . . . . . . 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 7450 | . . . . . . . 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 6909 | . . . . . . 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 6907 | . . . . . 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 7448 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑁(.g‘(mulGrp‘(Poly1‘𝐾)))(𝐹‘(var1‘(ℤ/nℤ‘𝑁)))) = (𝑁(.g‘(mulGrp‘(Poly1‘𝐾)))(var1‘𝐾))) | 
| 196 | 195, 93 | oveq12d 7450 | . . . . . . . . . 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 6909 | . . . . . . . . 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 6907 | . . . . . . . 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 2736 | . . . . . . . . . . 11
⊢
(eval1‘𝐾) = (eval1‘𝐾) | 
| 200 | 199, 89, 18, 60, 73, 8, 21 | evl1vard 22342 | . . . . . . . . . . . 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 22350 | . . . . . . . . . . 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 22201 | . . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ CRing →
(Poly1‘𝐾)
∈ CRing) | 
| 203 | 8, 202 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 →
(Poly1‘𝐾)
∈ CRing) | 
| 204 | 203 | crngringd 20244 | . . . . . . . . . . . . . 14
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Ring) | 
| 205 | 92 | zrhrhm 21523 | . . . . . . . . . . . . . . 15
⊢
((Poly1‘𝐾) ∈ Ring →
(ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring RingHom
(Poly1‘𝐾))) | 
| 206 | 49, 73 | rhmf 20486 | . . . . . . . . . . . . . . 15
⊢
((ℤRHom‘(Poly1‘𝐾)) ∈ (ℤring 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 7104 | . . . . . . . . . . . 12
⊢ (𝜑 →
((ℤRHom‘(Poly1‘𝐾))‘𝐴) ∈
(Base‘(Poly1‘𝐾))) | 
| 210 |  | eqidd 2737 | . . . . . . . . . . . 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 2736 | . . . . . . . . . . 11
⊢
(+g‘𝐾) = (+g‘𝐾) | 
| 213 | 199, 60, 18, 73, 8, 21, 201, 211, 86, 212 | evl1addd 22346 | . . . . . . . . . 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 21523 | . . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ Ring →
(ℤRHom‘𝐾)
∈ (ℤring RingHom 𝐾)) | 
| 216 | 49, 18 | rhmf 20486 | . . . . . . . . . . . . . . . . 17
⊢
((ℤRHom‘𝐾) ∈ (ℤring 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 7104 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) | 
| 220 | 199, 60, 18, 95, 73, 8, 219, 21 | evl1scad 22340 | . . . . . . . . . . . . 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 2742 | . . . . . . . . . . 11
⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (((eval1‘𝐾)‘((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀)) | 
| 223 | 97 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ (𝜑 →
((eval1‘𝐾)‘((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))) = ((eval1‘𝐾)‘((ℤRHom‘(Poly1‘𝐾))‘𝐴))) | 
| 224 | 223 | fveq1d 6907 | . . . . . . . . . . 11
⊢ (𝜑 →
(((eval1‘𝐾)‘((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴)))‘𝑀) = (((eval1‘𝐾)‘((ℤRHom‘(Poly1‘𝐾))‘𝐴))‘𝑀)) | 
| 225 | 222, 224 | eqtr2d 2777 | . . . . . . . . . 10
⊢ (𝜑 →
(((eval1‘𝐾)‘((ℤRHom‘(Poly1‘𝐾))‘𝐴))‘𝑀) = ((ℤRHom‘𝐾)‘𝐴)) | 
| 226 | 225 | oveq2d 7448 | . . . . . . . . 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 2776 | . . . . . . . 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 2776 | . . . . . . 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 19823 | . . . . . . . . . 10
⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd) | 
| 230 | 19, 14, 229, 27, 21 | mulgnn0cld 19114 | . . . . . . . . 9
⊢ (𝜑 → (𝑁(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾)) | 
| 231 | 199, 89, 18, 60, 73, 8, 230 | evl1vard 22342 | . . . . . . . . 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 22340 | . . . . . . . . 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 22346 | . . . . . . . 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 2779 | . . . . . 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 2776 | . . . . 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 2776 | . . . 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 2776 | . . 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 2776 | . 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 2780 | 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))) |