| Step | Hyp | Ref
| Expression |
| 1 | | aks6d1c6lem4.1 |
. 2
⊢ ∼ =
{〈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)))} |
| 2 | | aks6d1c6lem4.2 |
. 2
⊢ 𝑃 = (chr‘𝐾) |
| 3 | | aks6d1c6lem4.3 |
. 2
⊢ (𝜑 → 𝐾 ∈ Field) |
| 4 | | aks6d1c6lem4.4 |
. 2
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 5 | | aks6d1c6lem4.5 |
. 2
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 6 | | aks6d1c6lem4.6 |
. 2
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 7 | | aks6d1c6lem4.7 |
. 2
⊢ (𝜑 → 𝑃 ∥ 𝑁) |
| 8 | | aks6d1c6lem4.8 |
. 2
⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) |
| 9 | | simpr 484 |
. . 3
⊢ ((𝜑 ∧ 𝐴 < 𝑃) → 𝐴 < 𝑃) |
| 10 | | prmnn 16711 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 11 | 4, 10 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ ℕ) |
| 12 | 11 | nnred 12281 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ ℝ) |
| 13 | | aks6d1c6lem4.11 |
. . . . . . . . 9
⊢ 𝐴 =
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) |
| 14 | 5 | phicld 16809 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ϕ‘𝑅) ∈
ℕ) |
| 15 | 14 | nnred 12281 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (ϕ‘𝑅) ∈
ℝ) |
| 16 | 14 | nnnn0d 12587 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ϕ‘𝑅) ∈
ℕ0) |
| 17 | 16 | nn0ge0d 12590 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 ≤ (ϕ‘𝑅)) |
| 18 | 15, 17 | resqrtcld 15456 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
(√‘(ϕ‘𝑅)) ∈ ℝ) |
| 19 | | 2re 12340 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 2 ∈
ℝ) |
| 21 | | 2pos 12369 |
. . . . . . . . . . . . . . 15
⊢ 0 <
2 |
| 22 | 21 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 2) |
| 23 | 6 | nnred 12281 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 24 | 6 | nngt0d 12315 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 𝑁) |
| 25 | | 1red 11262 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 1 ∈
ℝ) |
| 26 | | 1lt2 12437 |
. . . . . . . . . . . . . . . . 17
⊢ 1 <
2 |
| 27 | 26 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 1 < 2) |
| 28 | 25, 27 | ltned 11397 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ≠ 2) |
| 29 | 28 | necomd 2996 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 2 ≠ 1) |
| 30 | 20, 22, 23, 24, 29 | relogbcld 41974 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 logb 𝑁) ∈
ℝ) |
| 31 | 18, 30 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈
ℝ) |
| 32 | 31 | flcld 13838 |
. . . . . . . . . . 11
⊢ (𝜑 →
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈
ℤ) |
| 33 | 15, 17 | sqrtge0d 15459 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ≤
(√‘(ϕ‘𝑅))) |
| 34 | 20 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 2 ∈
ℂ) |
| 35 | 22 | gt0ne0d 11827 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 2 ≠ 0) |
| 36 | | logb1 26812 |
. . . . . . . . . . . . . . . 16
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) =
0) |
| 37 | 34, 35, 29, 36 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2 logb 1) =
0) |
| 38 | 37 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 = (2 logb
1)) |
| 39 | | 2z 12649 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℤ |
| 40 | 39 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 2 ∈
ℤ) |
| 41 | 20 | leidd 11829 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 2 ≤ 2) |
| 42 | | 0lt1 11785 |
. . . . . . . . . . . . . . . 16
⊢ 0 <
1 |
| 43 | 42 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 0 < 1) |
| 44 | 6 | nnge1d 12314 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ≤ 𝑁) |
| 45 | 40, 41, 25, 43, 23, 24, 44 | logblebd 41977 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2 logb 1) ≤
(2 logb 𝑁)) |
| 46 | 38, 45 | eqbrtrd 5165 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ≤ (2 logb
𝑁)) |
| 47 | 18, 30, 33, 46 | mulge0d 11840 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤
((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) |
| 48 | | 0zd 12625 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℤ) |
| 49 | | flge 13845 |
. . . . . . . . . . . . 13
⊢
((((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ ∧ 0 ∈
ℤ) → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))) |
| 50 | 31, 48, 49 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0 ≤
((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))) |
| 51 | 47, 50 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ≤
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))) |
| 52 | 32, 51 | jca 511 |
. . . . . . . . . 10
⊢ (𝜑 →
((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))) |
| 53 | | elnn0z 12626 |
. . . . . . . . . 10
⊢
((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0
↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))) |
| 54 | 52, 53 | sylibr 234 |
. . . . . . . . 9
⊢ (𝜑 →
(⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈
ℕ0) |
| 55 | 13, 54 | eqeltrid 2845 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℕ0) |
| 56 | 55 | nn0red 12588 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 57 | 12, 56 | lenltd 11407 |
. . . . . 6
⊢ (𝜑 → (𝑃 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑃)) |
| 58 | 57 | biimpar 477 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃 ≤ 𝐴) |
| 59 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁)) |
| 60 | 59 | eqeq1d 2739 |
. . . . . . . 8
⊢ (𝑏 = 𝑃 → ((𝑏 gcd 𝑁) = 1 ↔ (𝑃 gcd 𝑁) = 1)) |
| 61 | | aks6d1c6lem4.9 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1) |
| 62 | 61 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1) |
| 63 | | 1zzd 12648 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ∈ ℤ) |
| 64 | 13, 32 | eqeltrid 2845 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 65 | 64 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝐴 ∈ ℤ) |
| 66 | 11 | nnzd 12640 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ ℤ) |
| 67 | 66 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ ℤ) |
| 68 | 11 | nnge1d 12314 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ≤ 𝑃) |
| 69 | 68 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ≤ 𝑃) |
| 70 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ≤ 𝐴) |
| 71 | 63, 65, 67, 69, 70 | elfzd 13555 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ (1...𝐴)) |
| 72 | 60, 62, 71 | rspcdva 3623 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → (𝑃 gcd 𝑁) = 1) |
| 73 | 72 | ex 412 |
. . . . . 6
⊢ (𝜑 → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1)) |
| 74 | 73 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1)) |
| 75 | 58, 74 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) = 1) |
| 76 | 6 | nnzd 12640 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 77 | | coprm 16748 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) |
| 78 | 4, 76, 77 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) |
| 79 | 78 | con1bid 355 |
. . . . . . . . . 10
⊢ (𝜑 → (¬ (𝑃 gcd 𝑁) = 1 ↔ 𝑃 ∥ 𝑁)) |
| 80 | 79 | bicomd 223 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ ¬ (𝑃 gcd 𝑁) = 1)) |
| 81 | 80 | biimpd 229 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∥ 𝑁 → ¬ (𝑃 gcd 𝑁) = 1)) |
| 82 | 7, 81 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝑃 gcd 𝑁) = 1) |
| 83 | 82 | neqned 2947 |
. . . . . 6
⊢ (𝜑 → (𝑃 gcd 𝑁) ≠ 1) |
| 84 | 83 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1) |
| 85 | 84 | neneqd 2945 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → ¬ (𝑃 gcd 𝑁) = 1) |
| 86 | 75, 85 | pm2.21dd 195 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝐴 < 𝑃) |
| 87 | 9, 86 | pm2.61dan 813 |
. 2
⊢ (𝜑 → 𝐴 < 𝑃) |
| 88 | | aks6d1c6lem4.10 |
. 2
⊢ 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))))))) |
| 89 | | aksaks6dlem4.12 |
. 2
⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
| 90 | | aks6d1c6lem4.13 |
. 2
⊢ 𝐿 =
(ℤRHom‘(ℤ/nℤ‘𝑅)) |
| 91 | | aks6d1c6lem4.14 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) |
| 92 | | aks6d1c6lem4.15 |
. 2
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) |
| 93 | | aks6d1c6lem4.16 |
. 2
⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) |
| 94 | | aks6d1c6lem4.17 |
. 2
⊢ 𝐻 = (ℎ ∈ (ℕ0
↑m (0...𝐴))
↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) |
| 95 | | aks6d1c6lem4.18 |
. 2
⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) |
| 96 | | aks6d1c6lem4.19 |
. 2
⊢ 𝑆 = {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} |
| 97 | | eqid 2737 |
. 2
⊢ (𝑗 ∈ (ℕ0
× ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) |
| 98 | | aks6d1c6lem4.21 |
. . 3
⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 ×
ℕ0))))) |
| 99 | | imaco 6271 |
. . . . . 6
⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 ×
ℕ0)) = (𝐽
“ (𝐸 “
(ℕ0 × ℕ0))) |
| 100 | 99 | eqcomi 2746 |
. . . . 5
⊢ (𝐽 “ (𝐸 “ (ℕ0 ×
ℕ0))) = ((𝐽 ∘ 𝐸) “ (ℕ0 ×
ℕ0)) |
| 101 | | resima 6033 |
. . . . . . . 8
⊢ (((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) “ (ℕ0 ×
ℕ0)) = ((𝐽
∘ 𝐸) “
(ℕ0 × ℕ0)) |
| 102 | 101 | eqcomi 2746 |
. . . . . . 7
⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 ×
ℕ0)) = (((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) “ (ℕ0 ×
ℕ0)) |
| 103 | 102 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 ×
ℕ0)) = (((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) “ (ℕ0 ×
ℕ0))) |
| 104 | 66 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → 𝑃 ∈ ℤ) |
| 105 | | xp1st 8046 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ (ℕ0
× ℕ0) → (1st ‘𝑣) ∈
ℕ0) |
| 106 | 105 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → (1st ‘𝑣) ∈
ℕ0) |
| 107 | 104, 106 | zexpcld 14128 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → (𝑃↑(1st ‘𝑣)) ∈
ℤ) |
| 108 | 11 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑃 ≠ 0) |
| 109 | | dvdsval2 16293 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ)) |
| 110 | 66, 108, 76, 109 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ)) |
| 111 | 7, 110 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ) |
| 112 | 111 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → (𝑁 / 𝑃) ∈ ℤ) |
| 113 | | xp2nd 8047 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ (ℕ0
× ℕ0) → (2nd ‘𝑣) ∈
ℕ0) |
| 114 | 113 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → (2nd ‘𝑣) ∈
ℕ0) |
| 115 | 112, 114 | zexpcld 14128 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) ∈
ℤ) |
| 116 | 107, 115 | zmulcld 12728 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) ∈
ℤ) |
| 117 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑘 ∈ V |
| 118 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑙 ∈ V |
| 119 | 117, 118 | op1std 8024 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 〈𝑘, 𝑙〉 → (1st ‘𝑣) = 𝑘) |
| 120 | 119 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 〈𝑘, 𝑙〉 → (𝑃↑(1st ‘𝑣)) = (𝑃↑𝑘)) |
| 121 | 117, 118 | op2ndd 8025 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 〈𝑘, 𝑙〉 → (2nd ‘𝑣) = 𝑙) |
| 122 | 121 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 〈𝑘, 𝑙〉 → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙)) |
| 123 | 120, 122 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 〈𝑘, 𝑙〉 → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
| 124 | 123 | mpompt 7547 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ (ℕ0
× ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) |
| 125 | 89, 124 | eqtr4i 2768 |
. . . . . . . . . . 11
⊢ 𝐸 = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) |
| 126 | 125 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐸 = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))) |
| 127 | | aks6d1c6lem4.20 |
. . . . . . . . . . 11
⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 128 | 127 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 129 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑗 = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 130 | 116, 126,
128, 129 | fmptco 7149 |
. . . . . . . . 9
⊢ (𝜑 → (𝐽 ∘ 𝐸) = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 131 | 130 | reseq1d 5996 |
. . . . . . . 8
⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) = ((𝑣
∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st
‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 ×
ℕ0))) |
| 132 | | ssidd 4007 |
. . . . . . . . . 10
⊢ (𝜑 → (ℕ0
× ℕ0) ⊆ (ℕ0 ×
ℕ0)) |
| 133 | 132 | resmptd 6058 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 ×
ℕ0)) = (𝑣
∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st
‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 134 | 126, 116 | fvmpt2d 7029 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑣) = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) |
| 135 | 134 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 136 | 135 | mpteq2dva 5242 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 137 | 136 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 138 | | ovexd 7466 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V) |
| 139 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ (ℕ0
× ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 140 | 138, 139 | fmptd 7134 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 ×
ℕ0)⟶V) |
| 141 | | ffn 6736 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ (ℕ0
× ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 ×
ℕ0)⟶V → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 ×
ℕ0)) |
| 142 | 140, 141 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 ×
ℕ0)) |
| 143 | | ovexd 7466 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V) |
| 144 | 143, 97 | fmptd 7134 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 ×
ℕ0)⟶V) |
| 145 | | ffn 6736 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ (ℕ0
× ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 ×
ℕ0)⟶V → (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 ×
ℕ0)) |
| 146 | 144, 145 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 ×
ℕ0)) |
| 147 | | eqidd 2738 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))) |
| 148 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑣 = 𝑐) → 𝑣 = 𝑐) |
| 149 | 148 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸‘𝑣) = (𝐸‘𝑐)) |
| 150 | 149 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑣 = 𝑐) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 151 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → 𝑐 ∈ (ℕ0 ×
ℕ0)) |
| 152 | | ovexd 7466 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V) |
| 153 | 147, 150,
151, 152 | fvmptd 7023 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) |
| 154 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
((mulGrp‘𝐾)
↾s 𝑈) =
((mulGrp‘𝐾)
↾s 𝑈) |
| 155 | | aks6d1c6lem4.22 |
. . . . . . . . . . . . . . . 16
⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈
(Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))} |
| 156 | 155 | ssrab3 4082 |
. . . . . . . . . . . . . . 15
⊢ 𝑈 ⊆
(Base‘(mulGrp‘𝐾)) |
| 157 | 156 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ⊆ (Base‘(mulGrp‘𝐾))) |
| 158 | 157 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → 𝑈 ⊆ (Base‘(mulGrp‘𝐾))) |
| 159 | 3 | fldcrngd 20742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐾 ∈ CRing) |
| 160 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(mulGrp‘𝐾) =
(mulGrp‘𝐾) |
| 161 | 160 | crngmgp 20238 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐾 ∈ CRing →
(mulGrp‘𝐾) ∈
CMnd) |
| 162 | 159, 161 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd) |
| 163 | 162, 5, 155 | primrootsunit 42099 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)) |
| 164 | 163 | simpld 494 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅)) |
| 165 | 93, 164 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅)) |
| 166 | 163 | simprd 495 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel) |
| 167 | | ablcmn 19805 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((mulGrp‘𝐾)
↾s 𝑈)
∈ Abel → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd) |
| 168 | 166, 167 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd) |
| 169 | 5 | nnnn0d 12587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑅 ∈
ℕ0) |
| 170 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢
(.g‘((mulGrp‘𝐾) ↾s 𝑈)) =
(.g‘((mulGrp‘𝐾) ↾s 𝑈)) |
| 171 | 168, 169,
170 | isprimroot 42094 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤)))) |
| 172 | 171 | biimpd 229 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤)))) |
| 173 | 165, 172 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) =
(0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤))) |
| 174 | 173 | simp1d 1143 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈))) |
| 175 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾)) |
| 176 | 154, 175 | ressbas2 17283 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈 ⊆
(Base‘(mulGrp‘𝐾)) → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈))) |
| 177 | 157, 176 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈))) |
| 178 | 174, 177 | eleqtrrd 2844 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ 𝑈) |
| 179 | 178 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → 𝑀 ∈ 𝑈) |
| 180 | 6, 4, 7, 89 | aks6d1c2p1 42119 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸:(ℕ0 ×
ℕ0)⟶ℕ) |
| 181 | 180 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑐) ∈ ℕ) |
| 182 | 154, 158,
179, 181 | ressmulgnnd 19096 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀)) |
| 183 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) |
| 184 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑐) → 𝑗 = 𝑐) |
| 185 | 184 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸‘𝑗) = (𝐸‘𝑐)) |
| 186 | 185 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀)) |
| 187 | | ovexd 7466 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V) |
| 188 | 183, 186,
151, 187 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀)) |
| 189 | 188 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) = ((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐)) |
| 190 | 153, 182,
189 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 ×
ℕ0)) → ((𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐)) |
| 191 | 142, 146,
190 | eqfnfvd 7054 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) |
| 192 | 137, 191 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) |
| 193 | 133, 192 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝑣 ∈ (ℕ0 ×
ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 ×
ℕ0)) = (𝑗
∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) |
| 194 | 131, 193 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) = (𝑗
∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) |
| 195 | 194 | imaeq1d 6077 |
. . . . . 6
⊢ (𝜑 → (((𝐽 ∘ 𝐸) ↾ (ℕ0 ×
ℕ0)) “ (ℕ0 ×
ℕ0)) = ((𝑗
∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 ×
ℕ0))) |
| 196 | 103, 195 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 ×
ℕ0)) = ((𝑗
∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 ×
ℕ0))) |
| 197 | 100, 196 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 ×
ℕ0))) = ((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 ×
ℕ0))) |
| 198 | 197 | fveq2d 6910 |
. . 3
⊢ (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 ×
ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 ×
ℕ0)))) |
| 199 | 98, 198 | breqtrd 5169 |
. 2
⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) ≤ (♯‘((𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 ×
ℕ0)))) |
| 200 | 1, 2, 3, 4, 5, 6, 7, 8, 87, 88, 55, 89, 90, 91, 92, 93, 94, 95, 96, 97, 199 | aks6d1c6lem3 42173 |
1
⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0
↑m (0...𝐴))))) |