| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | aks6d1c2p3.4 | . . . 4
⊢ (𝜑 → 𝐽 = (𝑟𝐸𝑜)) | 
| 2 |  | aks6d1c2.12 | . . . . . 6
⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) | 
| 3 | 2 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))) | 
| 4 |  | simprl 770 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑘 = 𝑟) | 
| 5 | 4 | oveq2d 7448 | . . . . . 6
⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → (𝑃↑𝑘) = (𝑃↑𝑟)) | 
| 6 |  | simprr 772 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑙 = 𝑜) | 
| 7 | 6 | oveq2d 7448 | . . . . . 6
⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜)) | 
| 8 | 5, 7 | oveq12d 7450 | . . . . 5
⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))) | 
| 9 |  | aks6d1c2p3.2 | . . . . . 6
⊢ (𝜑 → 𝑟 ∈ (0...𝐵)) | 
| 10 |  | elfznn0 13661 | . . . . . 6
⊢ (𝑟 ∈ (0...𝐵) → 𝑟 ∈ ℕ0) | 
| 11 | 9, 10 | syl 17 | . . . . 5
⊢ (𝜑 → 𝑟 ∈ ℕ0) | 
| 12 |  | aks6d1c2p3.3 | . . . . . 6
⊢ (𝜑 → 𝑜 ∈ (0...𝐵)) | 
| 13 |  | elfznn0 13661 | . . . . . 6
⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ0) | 
| 14 | 12, 13 | syl 17 | . . . . 5
⊢ (𝜑 → 𝑜 ∈ ℕ0) | 
| 15 |  | ovexd 7467 | . . . . 5
⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V) | 
| 16 | 3, 8, 11, 14, 15 | ovmpod 7586 | . . . 4
⊢ (𝜑 → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))) | 
| 17 | 1, 16 | eqtrd 2776 | . . 3
⊢ (𝜑 → 𝐽 = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))) | 
| 18 | 17 | oveq1d 7447 | . 2
⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 19 |  | aks6d1c2p3.7 | . . . . 5
⊢ (𝜑 → 𝐼 = (𝑝𝐸𝑞)) | 
| 20 |  | simprl 770 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑘 = 𝑝) | 
| 21 | 20 | oveq2d 7448 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → (𝑃↑𝑘) = (𝑃↑𝑝)) | 
| 22 |  | simprr 772 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑙 = 𝑞) | 
| 23 | 22 | oveq2d 7448 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞)) | 
| 24 | 21, 23 | oveq12d 7450 | . . . . . 6
⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))) | 
| 25 |  | aks6d1c2p3.5 | . . . . . . 7
⊢ (𝜑 → 𝑝 ∈ (0...𝐵)) | 
| 26 |  | elfznn0 13661 | . . . . . . 7
⊢ (𝑝 ∈ (0...𝐵) → 𝑝 ∈ ℕ0) | 
| 27 | 25, 26 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝑝 ∈ ℕ0) | 
| 28 |  | aks6d1c2p3.6 | . . . . . . 7
⊢ (𝜑 → 𝑞 ∈ (0...𝐵)) | 
| 29 |  | elfznn0 13661 | . . . . . . 7
⊢ (𝑞 ∈ (0...𝐵) → 𝑞 ∈ ℕ0) | 
| 30 | 28, 29 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝑞 ∈ ℕ0) | 
| 31 |  | ovexd 7467 | . . . . . 6
⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V) | 
| 32 | 3, 24, 27, 30, 31 | ovmpod 7586 | . . . . 5
⊢ (𝜑 → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))) | 
| 33 | 19, 32 | eqtrd 2776 | . . . 4
⊢ (𝜑 → 𝐼 = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))) | 
| 34 | 33 | oveq1d 7447 | . . 3
⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 35 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) | 
| 36 | 35 | oveq2d 7448 | . . . . . 6
⊢ (𝑦 = 𝑀 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 37 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑦 = 𝑀 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)) | 
| 38 | 37 | fveq2d 6909 | . . . . . 6
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀))) | 
| 39 | 36, 38 | eqeq12d 2752 | . . . . 5
⊢ (𝑦 = 𝑀 → ((((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)) ↔ (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 40 |  | aks6d1c2.1 | . . . . . . 7
⊢  ∼ =
{〈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)))} | 
| 41 |  | aks6d1c2.2 | . . . . . . 7
⊢ 𝑃 = (chr‘𝐾) | 
| 42 |  | aks6d1c2.3 | . . . . . . 7
⊢ (𝜑 → 𝐾 ∈ Field) | 
| 43 |  | aks6d1c2.4 | . . . . . . 7
⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| 44 |  | aks6d1c2.5 | . . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℕ) | 
| 45 |  | aks6d1c2.6 | . . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 46 |  | aks6d1c2.7 | . . . . . . 7
⊢ (𝜑 → 𝑃 ∥ 𝑁) | 
| 47 |  | aks6d1c2.8 | . . . . . . 7
⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | 
| 48 |  | aks6d1c2p3.1 | . . . . . . . 8
⊢ (𝜑 → 𝑠 ∈ (ℕ0
↑m (0...𝐴))) | 
| 49 |  | nn0ex 12534 | . . . . . . . . . 10
⊢
ℕ0 ∈ V | 
| 50 | 49 | a1i 11 | . . . . . . . . 9
⊢ (𝜑 → ℕ0 ∈
V) | 
| 51 |  | ovexd 7467 | . . . . . . . . 9
⊢ (𝜑 → (0...𝐴) ∈ V) | 
| 52 |  | elmapg 8880 | . . . . . . . . 9
⊢
((ℕ0 ∈ V ∧ (0...𝐴) ∈ V) → (𝑠 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑠:(0...𝐴)⟶ℕ0)) | 
| 53 | 50, 51, 52 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝑠 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑠:(0...𝐴)⟶ℕ0)) | 
| 54 | 48, 53 | mpbid 232 | . . . . . . 7
⊢ (𝜑 → 𝑠:(0...𝐴)⟶ℕ0) | 
| 55 |  | aks6d1c2.10 | . . . . . . 7
⊢ 𝐺 = (𝑔 ∈ (ℕ0
↑m (0...𝐴))
↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) | 
| 56 |  | aks6d1c2.11 | . . . . . . 7
⊢ (𝜑 → 𝐴 ∈
ℕ0) | 
| 57 |  | eqid 2736 | . . . . . . 7
⊢ ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) | 
| 58 |  | aks6d1c2.14 | . . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) | 
| 59 |  | aks6d1c2.15 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) | 
| 60 | 40, 41, 42, 43, 44, 45, 46, 47, 54, 55, 56, 27, 30, 57, 58, 59 | aks6d1c1rh 42127 | . . . . . 6
⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∼ (𝐺‘𝑠)) | 
| 61 | 40, 60 | aks6d1c1p1rcl 42110 | . . . . . . . 8
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ ∧ (𝐺‘𝑠) ∈
(Base‘(Poly1‘𝐾)))) | 
| 62 | 61 | simprd 495 | . . . . . . 7
⊢ (𝜑 → (𝐺‘𝑠) ∈
(Base‘(Poly1‘𝐾))) | 
| 63 | 61 | simpld 494 | . . . . . . 7
⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ) | 
| 64 | 40, 62, 63 | aks6d1c1p1 42109 | . . . . . 6
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∼ (𝐺‘𝑠) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)))) | 
| 65 | 60, 64 | mpbid 232 | . . . . 5
⊢ (𝜑 → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦))) | 
| 66 |  | aks6d1c2.16 | . . . . 5
⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) | 
| 67 | 39, 65, 66 | rspcdva 3622 | . . . 4
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀))) | 
| 68 | 33 | eqcomd 2742 | . . . . . . . 8
⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) = 𝐼) | 
| 69 | 68 | oveq1d 7447 | . . . . . . 7
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 70 | 17 | eqcomd 2742 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) = 𝐽) | 
| 71 | 70 | oveq1d 7447 | . . . . . . . 8
⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀) = (𝐽(.g‘(mulGrp‘𝐾))𝑀)) | 
| 72 |  | aks6d1c2.27 | . . . . . . . . . 10
⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅))) | 
| 73 | 72 | oveq1d 7447 | . . . . . . . . 9
⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀)) | 
| 74 | 42 | fldcrngd 20743 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 ∈ CRing) | 
| 75 |  | crngring 20243 | . . . . . . . . . . . . 13
⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring) | 
| 76 | 74, 75 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ Ring) | 
| 77 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(mulGrp‘𝐾) =
(mulGrp‘𝐾) | 
| 78 | 77 | ringmgp 20237 | . . . . . . . . . . . 12
⊢ (𝐾 ∈ Ring →
(mulGrp‘𝐾) ∈
Mnd) | 
| 79 | 76, 78 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd) | 
| 80 |  | aks6d1c2p3.8 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ∈
ℕ0) | 
| 81 |  | aks6d1c2.26 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ ℕ) | 
| 82 | 81 | nnnn0d 12589 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑈 ∈
ℕ0) | 
| 83 | 44 | nnnn0d 12589 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈
ℕ0) | 
| 84 | 82, 83 | nn0mulcld 12594 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑈 · 𝑅) ∈
ℕ0) | 
| 85 | 77 | crngmgp 20239 | . . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ CRing →
(mulGrp‘𝐾) ∈
CMnd) | 
| 86 | 74, 85 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd) | 
| 87 |  | eqid 2736 | . . . . . . . . . . . . . . . 16
⊢
(.g‘(mulGrp‘𝐾)) =
(.g‘(mulGrp‘𝐾)) | 
| 88 | 86, 83, 87 | isprimroot 42095 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0
((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣)))) | 
| 89 | 88 | biimpd 229 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0
((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣)))) | 
| 90 | 66, 89 | mpd 15 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0
((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣))) | 
| 91 | 90 | simp1d 1142 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾))) | 
| 92 | 80, 84, 91 | 3jca 1128 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐼 ∈ ℕ0 ∧ (𝑈 · 𝑅) ∈ ℕ0 ∧ 𝑀 ∈
(Base‘(mulGrp‘𝐾)))) | 
| 93 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾)) | 
| 94 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(+g‘(mulGrp‘𝐾)) =
(+g‘(mulGrp‘𝐾)) | 
| 95 | 93, 87, 94 | mulgnn0dir 19123 | . . . . . . . . . . 11
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ (𝐼 ∈
ℕ0 ∧ (𝑈 · 𝑅) ∈ ℕ0 ∧ 𝑀 ∈
(Base‘(mulGrp‘𝐾)))) → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 96 | 79, 92, 95 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 97 | 82, 83, 91 | 3jca 1128 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0
∧ 𝑀 ∈
(Base‘(mulGrp‘𝐾)))) | 
| 98 | 93, 87 | mulgnn0ass 19129 | . . . . . . . . . . . . . 14
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ (𝑈 ∈
ℕ0 ∧ 𝑅
∈ ℕ0 ∧ 𝑀 ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀))) | 
| 99 | 79, 97, 98 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀))) | 
| 100 | 90 | simp2d 1143 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾))) | 
| 101 | 100 | oveq2d 7448 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)) = (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾)))) | 
| 102 |  | eqid 2736 | . . . . . . . . . . . . . . . 16
⊢
(0g‘(mulGrp‘𝐾)) =
(0g‘(mulGrp‘𝐾)) | 
| 103 | 93, 87, 102 | mulgnn0z 19120 | . . . . . . . . . . . . . . 15
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ 𝑈 ∈
ℕ0) → (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) =
(0g‘(mulGrp‘𝐾))) | 
| 104 | 79, 82, 103 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) =
(0g‘(mulGrp‘𝐾))) | 
| 105 | 101, 104 | eqtrd 2776 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)) =
(0g‘(mulGrp‘𝐾))) | 
| 106 | 99, 105 | eqtrd 2776 | . . . . . . . . . . . 12
⊢ (𝜑 → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾))) | 
| 107 | 106 | oveq2d 7448 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾)))) | 
| 108 | 93, 87, 79, 80, 91 | mulgnn0cld 19114 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘(mulGrp‘𝐾))) | 
| 109 | 93, 94, 102 | mndrid 18769 | . . . . . . . . . . . 12
⊢
(((mulGrp‘𝐾)
∈ Mnd ∧ (𝐼(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘(mulGrp‘𝐾))) → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 110 | 79, 108, 109 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 111 | 107, 110 | eqtrd 2776 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 112 | 96, 111 | eqtrd 2776 | . . . . . . . . 9
⊢ (𝜑 → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 113 | 73, 112 | eqtrd 2776 | . . . . . . . 8
⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀)) | 
| 114 | 71, 113 | eqtr2d 2777 | . . . . . . 7
⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))𝑀) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)) | 
| 115 | 69, 114 | eqtrd 2776 | . . . . . 6
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)) | 
| 116 | 115 | fveq2d 6909 | . . . . 5
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))) | 
| 117 | 35 | oveq2d 7448 | . . . . . . . 8
⊢ (𝑦 = 𝑀 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 118 |  | oveq2 7440 | . . . . . . . . 9
⊢ (𝑦 = 𝑀 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)) | 
| 119 | 118 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))) | 
| 120 | 117, 119 | eqeq12d 2752 | . . . . . . 7
⊢ (𝑦 = 𝑀 → ((((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)) ↔ (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 121 |  | eqid 2736 | . . . . . . . . 9
⊢ ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) | 
| 122 | 40, 41, 42, 43, 44, 45, 46, 47, 54, 55, 56, 11, 14, 121, 58, 59 | aks6d1c1rh 42127 | . . . . . . . 8
⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∼ (𝐺‘𝑠)) | 
| 123 | 40, 122 | aks6d1c1p1rcl 42110 | . . . . . . . . . 10
⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ ∧ (𝐺‘𝑠) ∈
(Base‘(Poly1‘𝐾)))) | 
| 124 | 123 | simpld 494 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ) | 
| 125 | 40, 62, 124 | aks6d1c1p1 42109 | . . . . . . . 8
⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∼ (𝐺‘𝑠) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)))) | 
| 126 | 122, 125 | mpbid 232 | . . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦))) | 
| 127 | 120, 126,
66 | rspcdva 3622 | . . . . . 6
⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))) | 
| 128 | 127 | eqcomd 2742 | . . . . 5
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 129 | 116, 128 | eqtrd 2776 | . . . 4
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 130 | 67, 129 | eqtrd 2776 | . . 3
⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 131 | 34, 130 | eqtr2d 2777 | . 2
⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) | 
| 132 | 18, 131 | eqtrd 2776 | 1
⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))) |