| Step | Hyp | Ref
| Expression |
| 1 | | aks4d1p7.1 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘3)) |
| 2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) → 𝑁 ∈
(ℤ≥‘3)) |
| 3 | | aks4d1p7.2 |
. . . . 5
⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) |
| 4 | | aks4d1p7.3 |
. . . . 5
⊢ 𝐵 = (⌈‘((2
logb 𝑁)↑5)) |
| 5 | | aks4d1p7.4 |
. . . . 5
⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) |
| 6 | | breq1 5146 |
. . . . . . . . 9
⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝑅 ↔ 𝑞 ∥ 𝑅)) |
| 7 | | breq1 5146 |
. . . . . . . . 9
⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝑁 ↔ 𝑞 ∥ 𝑁)) |
| 8 | 6, 7 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑝 = 𝑞 → ((𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁) ↔ (𝑞 ∥ 𝑅 → 𝑞 ∥ 𝑁))) |
| 9 | 8 | cbvralvw 3237 |
. . . . . . 7
⊢
(∀𝑝 ∈
ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁) ↔ ∀𝑞 ∈ ℙ (𝑞 ∥ 𝑅 → 𝑞 ∥ 𝑁)) |
| 10 | 9 | biimpi 216 |
. . . . . 6
⊢
(∀𝑝 ∈
ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁) → ∀𝑞 ∈ ℙ (𝑞 ∥ 𝑅 → 𝑞 ∥ 𝑁)) |
| 11 | 10 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) → ∀𝑞 ∈ ℙ (𝑞 ∥ 𝑅 → 𝑞 ∥ 𝑁)) |
| 12 | 2, 3, 4, 5, 11 | aks4d1p7d1 42083 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) → 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) |
| 13 | 5 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )) |
| 14 | | ltso 11341 |
. . . . . . . . . . 11
⊢ < Or
ℝ |
| 15 | 14 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → < Or
ℝ) |
| 16 | | fzfid 14014 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1...𝐵) ∈ Fin) |
| 17 | | ssrab2 4080 |
. . . . . . . . . . . . 13
⊢ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵) |
| 18 | 17 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ (1...𝐵)) |
| 19 | 16, 18 | ssfid 9301 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin) |
| 20 | 1, 3, 4 | aks4d1p3 42079 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴) |
| 21 | | rabn0 4389 |
. . . . . . . . . . . 12
⊢ ({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ↔ ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴) |
| 22 | 20, 21 | sylibr 234 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅) |
| 23 | | elfznn 13593 |
. . . . . . . . . . . . . . . 16
⊢ (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℕ) |
| 24 | 23 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℕ) |
| 25 | 24 | nnred 12281 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑜 ∈ (1...𝐵)) → 𝑜 ∈ ℝ) |
| 26 | 25 | ex 412 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑜 ∈ (1...𝐵) → 𝑜 ∈ ℝ)) |
| 27 | 26 | ssrdv 3989 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1...𝐵) ⊆ ℝ) |
| 28 | 18, 27 | sstrd 3994 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ) |
| 29 | 19, 22, 28 | 3jca 1129 |
. . . . . . . . . 10
⊢ (𝜑 → ({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)) |
| 30 | | fiinfcl 9541 |
. . . . . . . . . 10
⊢ (( <
Or ℝ ∧ ({𝑟 ∈
(1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ∈ Fin ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ≠ ∅ ∧ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ⊆ ℝ)) → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}) |
| 31 | 15, 29, 30 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}) |
| 32 | 13, 31 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}) |
| 33 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (𝑟 ∥ 𝐴 ↔ 𝑅 ∥ 𝐴)) |
| 34 | 33 | notbid 318 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (¬ 𝑟 ∥ 𝐴 ↔ ¬ 𝑅 ∥ 𝐴)) |
| 35 | 34 | elrab 3692 |
. . . . . . . 8
⊢ (𝑅 ∈ {𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴} ↔ (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴)) |
| 36 | 32, 35 | sylib 218 |
. . . . . . 7
⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴)) |
| 37 | 36 | simprd 495 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑅 ∥ 𝐴) |
| 38 | 1, 3, 4, 5 | aks4d1p4 42080 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴)) |
| 39 | 38 | simpld 494 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ∈ (1...𝐵)) |
| 40 | 39 | elfzelzd 13565 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ ℤ) |
| 41 | | eluzelz 12888 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘3) → 𝑁 ∈ ℤ) |
| 42 | 1, 41 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 43 | | 2re 12340 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ |
| 44 | 43 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 2 ∈
ℝ) |
| 45 | | 2pos 12369 |
. . . . . . . . . . . . . . . . 17
⊢ 0 <
2 |
| 46 | 45 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 < 2) |
| 47 | 4 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 = (⌈‘((2 logb 𝑁)↑5))) |
| 48 | 42 | zred 12722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 49 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 0 ∈
ℝ) |
| 50 | | 3re 12346 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 3 ∈
ℝ |
| 51 | 50 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 3 ∈
ℝ) |
| 52 | | 3pos 12371 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 <
3 |
| 53 | 52 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 0 < 3) |
| 54 | | eluzle 12891 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈
(ℤ≥‘3) → 3 ≤ 𝑁) |
| 55 | 1, 54 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 3 ≤ 𝑁) |
| 56 | 49, 51, 48, 53, 55 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 < 𝑁) |
| 57 | | 1red 11262 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 ∈
ℝ) |
| 58 | | 1lt2 12437 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 1 <
2 |
| 59 | 58 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 < 2) |
| 60 | 57, 59 | ltned 11397 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 1 ≠ 2) |
| 61 | 60 | necomd 2996 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 2 ≠ 1) |
| 62 | 44, 46, 48, 56, 61 | relogbcld 41974 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (2 logb 𝑁) ∈
ℝ) |
| 63 | | 5nn0 12546 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 5 ∈
ℕ0 |
| 64 | 63 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 5 ∈
ℕ0) |
| 65 | 62, 64 | reexpcld 14203 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈
ℝ) |
| 66 | 65 | ceilcld 13883 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (⌈‘((2
logb 𝑁)↑5))
∈ ℤ) |
| 67 | 66 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (⌈‘((2
logb 𝑁)↑5))
∈ ℝ) |
| 68 | 47, 67 | eqeltrd 2841 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 69 | | 9re 12365 |
. . . . . . . . . . . . . . . . . 18
⊢ 9 ∈
ℝ |
| 70 | 69 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 9 ∈
ℝ) |
| 71 | | 9pos 12379 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 <
9 |
| 72 | 71 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 < 9) |
| 73 | 48, 55 | 3lexlogpow5ineq4 42057 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 9 < ((2 logb
𝑁)↑5)) |
| 74 | 65 | ceilged 13886 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2 logb 𝑁)↑5) ≤
(⌈‘((2 logb 𝑁)↑5))) |
| 75 | 70, 65, 67, 73, 74 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 9 < (⌈‘((2
logb 𝑁)↑5))) |
| 76 | 75, 47 | breqtrrd 5171 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 9 < 𝐵) |
| 77 | 49, 70, 68, 72, 76 | lttrd 11422 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 < 𝐵) |
| 78 | 44, 46, 68, 77, 61 | relogbcld 41974 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2 logb 𝐵) ∈
ℝ) |
| 79 | 78 | flcld 13838 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (⌊‘(2
logb 𝐵)) ∈
ℤ) |
| 80 | 44 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 2 ∈
ℂ) |
| 81 | 49, 46 | gtned 11396 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 2 ≠ 0) |
| 82 | | logb1 26812 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) =
0) |
| 83 | 80, 81, 61, 82 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2 logb 1) =
0) |
| 84 | 83 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 = (2 logb
1)) |
| 85 | | 2z 12649 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℤ |
| 86 | 85 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ∈
ℤ) |
| 87 | 44 | leidd 11829 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2 ≤ 2) |
| 88 | | 0lt1 11785 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 <
1 |
| 89 | 88 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 < 1) |
| 90 | | 1lt9 12472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 1 <
9 |
| 91 | 90 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 1 < 9) |
| 92 | 57, 70, 91 | ltled 11409 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 1 ≤ 9) |
| 93 | 70, 68, 76 | ltled 11409 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 9 ≤ 𝐵) |
| 94 | 57, 70, 68, 92, 93 | letrd 11418 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ≤ 𝐵) |
| 95 | 86, 87, 57, 89, 68, 77, 94 | logblebd 41977 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2 logb 1) ≤
(2 logb 𝐵)) |
| 96 | 84, 95 | eqbrtrd 5165 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 0 ≤ (2 logb
𝐵)) |
| 97 | | 0zd 12625 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ∈
ℤ) |
| 98 | | flge 13845 |
. . . . . . . . . . . . . . . 16
⊢ (((2
logb 𝐵) ∈
ℝ ∧ 0 ∈ ℤ) → (0 ≤ (2 logb 𝐵) ↔ 0 ≤
(⌊‘(2 logb 𝐵)))) |
| 99 | 78, 97, 98 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0 ≤ (2 logb
𝐵) ↔ 0 ≤
(⌊‘(2 logb 𝐵)))) |
| 100 | 96, 99 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 ≤ (⌊‘(2
logb 𝐵))) |
| 101 | 79, 100 | jca 511 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((⌊‘(2
logb 𝐵)) ∈
ℤ ∧ 0 ≤ (⌊‘(2 logb 𝐵)))) |
| 102 | | elnn0z 12626 |
. . . . . . . . . . . . 13
⊢
((⌊‘(2 logb 𝐵)) ∈ ℕ0 ↔
((⌊‘(2 logb 𝐵)) ∈ ℤ ∧ 0 ≤
(⌊‘(2 logb 𝐵)))) |
| 103 | 101, 102 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (𝜑 → (⌊‘(2
logb 𝐵)) ∈
ℕ0) |
| 104 | 42, 103 | zexpcld 14128 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁↑(⌊‘(2 logb
𝐵))) ∈
ℤ) |
| 105 | | fzfid 14014 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1...(⌊‘((2
logb 𝑁)↑2))) ∈ Fin) |
| 106 | 42 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → 𝑁 ∈ ℤ) |
| 107 | | elfznn 13593 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2))) → 𝑘 ∈ ℕ) |
| 108 | 107 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → 𝑘 ∈ ℕ) |
| 109 | 108 | nnnn0d 12587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → 𝑘 ∈ ℕ0) |
| 110 | 106, 109 | zexpcld 14128 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ) |
| 111 | | 1zzd 12648 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → 1 ∈
ℤ) |
| 112 | 110, 111 | zsubcld 12727 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ) |
| 113 | 105, 112 | fprodzcl 15990 |
. . . . . . . . . . 11
⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ) |
| 114 | | dvdsmultr1 16333 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ ℤ ∧ (𝑁↑(⌊‘(2
logb 𝐵))) ∈
ℤ ∧ ∏𝑘
∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ) → (𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵))) → 𝑅 ∥ ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1)))) |
| 115 | 40, 104, 113, 114 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵))) → 𝑅 ∥ ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1)))) |
| 116 | 115 | imp 406 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) → 𝑅 ∥ ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1))) |
| 117 | 3 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 = ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1))) |
| 118 | 117 | breq2d 5155 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ∥ 𝐴 ↔ 𝑅 ∥ ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1)))) |
| 119 | 118 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) → (𝑅 ∥ 𝐴 ↔ 𝑅 ∥ ((𝑁↑(⌊‘(2 logb
𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2
logb 𝑁)↑2)))((𝑁↑𝑘) − 1)))) |
| 120 | 116, 119 | mpbird 257 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) → 𝑅 ∥ 𝐴) |
| 121 | 120 | ex 412 |
. . . . . . 7
⊢ (𝜑 → (𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵))) → 𝑅 ∥ 𝐴)) |
| 122 | 121 | con3d 152 |
. . . . . 6
⊢ (𝜑 → (¬ 𝑅 ∥ 𝐴 → ¬ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵))))) |
| 123 | 37, 122 | mpd 15 |
. . . . 5
⊢ (𝜑 → ¬ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) |
| 124 | 123 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) → ¬ 𝑅 ∥ (𝑁↑(⌊‘(2 logb
𝐵)))) |
| 125 | 12, 124 | pm2.65da 817 |
. . 3
⊢ (𝜑 → ¬ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 126 | | ianor 984 |
. . . . . . . 8
⊢ (¬
(𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ (¬ 𝑝 ∥ 𝑅 ∨ ¬ ¬ 𝑝 ∥ 𝑁)) |
| 127 | | notnotb 315 |
. . . . . . . . . 10
⊢ (𝑝 ∥ 𝑁 ↔ ¬ ¬ 𝑝 ∥ 𝑁) |
| 128 | 127 | orbi2i 913 |
. . . . . . . . 9
⊢ ((¬
𝑝 ∥ 𝑅 ∨ 𝑝 ∥ 𝑁) ↔ (¬ 𝑝 ∥ 𝑅 ∨ ¬ ¬ 𝑝 ∥ 𝑁)) |
| 129 | 128 | bicomi 224 |
. . . . . . . 8
⊢ ((¬
𝑝 ∥ 𝑅 ∨ ¬ ¬ 𝑝 ∥ 𝑁) ↔ (¬ 𝑝 ∥ 𝑅 ∨ 𝑝 ∥ 𝑁)) |
| 130 | 126, 129 | bitri 275 |
. . . . . . 7
⊢ (¬
(𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ (¬ 𝑝 ∥ 𝑅 ∨ 𝑝 ∥ 𝑁)) |
| 131 | | df-or 849 |
. . . . . . 7
⊢ ((¬
𝑝 ∥ 𝑅 ∨ 𝑝 ∥ 𝑁) ↔ (¬ ¬ 𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 132 | 130, 131 | bitri 275 |
. . . . . 6
⊢ (¬
(𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ (¬ ¬ 𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 133 | | notnotb 315 |
. . . . . . . 8
⊢ (𝑝 ∥ 𝑅 ↔ ¬ ¬ 𝑝 ∥ 𝑅) |
| 134 | 133 | imbi1i 349 |
. . . . . . 7
⊢ ((𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁) ↔ (¬ ¬ 𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 135 | 134 | bicomi 224 |
. . . . . 6
⊢ ((¬
¬ 𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁) ↔ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 136 | 132, 135 | bitri 275 |
. . . . 5
⊢ (¬
(𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 137 | 136 | ralbii 3093 |
. . . 4
⊢
(∀𝑝 ∈
ℙ ¬ (𝑝 ∥
𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 138 | 137 | notbii 320 |
. . 3
⊢ (¬
∀𝑝 ∈ ℙ
¬ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) |
| 139 | 125, 138 | sylibr 234 |
. 2
⊢ (𝜑 → ¬ ∀𝑝 ∈ ℙ ¬ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) |
| 140 | | ralnex 3072 |
. . . 4
⊢
(∀𝑝 ∈
ℙ ¬ (𝑝 ∥
𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ ¬ ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) |
| 141 | 140 | con2bii 357 |
. . 3
⊢
(∃𝑝 ∈
ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ ¬ ∀𝑝 ∈ ℙ ¬ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) |
| 142 | 141 | bicomi 224 |
. 2
⊢ (¬
∀𝑝 ∈ ℙ
¬ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁) ↔ ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) |
| 143 | 139, 142 | sylib 218 |
1
⊢ (𝜑 → ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) |