| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 𝑁 ∈
ℕ0) |
| 2 | | nn0nnaddcl 12557 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (𝑁 + 𝑀) ∈
ℕ) |
| 3 | | nnm1nn0 12567 |
. . . . 5
⊢ ((𝑁 + 𝑀) ∈ ℕ → ((𝑁 + 𝑀) − 1) ∈
ℕ0) |
| 4 | 2, 3 | syl 17 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ ((𝑁 + 𝑀) − 1) ∈
ℕ0) |
| 5 | | 1red 11262 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 1 ∈ ℝ) |
| 6 | | nnre 12273 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℝ) |
| 7 | 6 | adantl 481 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 𝑀 ∈
ℝ) |
| 8 | | nn0re 12535 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 𝑁 ∈
ℝ) |
| 10 | | nnge1 12294 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 1 ≤
𝑀) |
| 11 | 10 | adantl 481 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 1 ≤ 𝑀) |
| 12 | 5, 7, 9, 11 | leadd2dd 11878 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (𝑁 + 1) ≤ (𝑁 + 𝑀)) |
| 13 | | readdcl 11238 |
. . . . . . 7
⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 + 𝑀) ∈ ℝ) |
| 14 | 8, 6, 13 | syl2an 596 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (𝑁 + 𝑀) ∈
ℝ) |
| 15 | | leaddsub 11739 |
. . . . . 6
⊢ ((𝑁 ∈ ℝ ∧ 1 ∈
ℝ ∧ (𝑁 + 𝑀) ∈ ℝ) → ((𝑁 + 1) ≤ (𝑁 + 𝑀) ↔ 𝑁 ≤ ((𝑁 + 𝑀) − 1))) |
| 16 | 9, 5, 14, 15 | syl3anc 1373 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ ((𝑁 + 1) ≤ (𝑁 + 𝑀) ↔ 𝑁 ≤ ((𝑁 + 𝑀) − 1))) |
| 17 | 12, 16 | mpbid 232 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 𝑁 ≤ ((𝑁 + 𝑀) − 1)) |
| 18 | | elfz2nn0 13658 |
. . . 4
⊢ (𝑁 ∈ (0...((𝑁 + 𝑀) − 1)) ↔ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 𝑀) − 1) ∈ ℕ0
∧ 𝑁 ≤ ((𝑁 + 𝑀) − 1))) |
| 19 | 1, 4, 17, 18 | syl3anbrc 1344 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 𝑁 ∈
(0...((𝑁 + 𝑀) − 1))) |
| 20 | | fzfid 14014 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (0...((𝑁 + 𝑀) − 1)) ∈
Fin) |
| 21 | | fz0ssnn0 13662 |
. . . . 5
⊢
(0...((𝑁 + 𝑀) − 1)) ⊆
ℕ0 |
| 22 | 21 | a1i 11 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (0...((𝑁 + 𝑀) − 1)) ⊆
ℕ0) |
| 23 | | 2nn0 12543 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
| 24 | 23 | a1i 11 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
→ 2 ∈ ℕ0) |
| 25 | | id 22 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℕ0) |
| 26 | 24, 25 | nn0expcld 14285 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
→ (2↑𝑛) ∈
ℕ0) |
| 27 | 24, 26 | nn0expcld 14285 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ0
→ (2↑(2↑𝑛))
∈ ℕ0) |
| 28 | 27 | nn0zd 12639 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
→ (2↑(2↑𝑛))
∈ ℤ) |
| 29 | 28 | peano2zd 12725 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ ((2↑(2↑𝑛))
+ 1) ∈ ℤ) |
| 30 | 29 | adantl 481 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
∧ 𝑛 ∈
ℕ0) → ((2↑(2↑𝑛)) + 1) ∈ ℤ) |
| 31 | | df-fmtno 47515 |
. . . . 5
⊢ FermatNo
= (𝑛 ∈
ℕ0 ↦ ((2↑(2↑𝑛)) + 1)) |
| 32 | 30, 31 | fmptd 7134 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ FermatNo:ℕ0⟶ℤ) |
| 33 | 20, 22, 32 | fprodfvdvdsd 16371 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ ∀𝑛 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑛)
∥ ∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘)) |
| 34 | | fveq2 6906 |
. . . . 5
⊢ (𝑛 = 𝑁 → (FermatNo‘𝑛) = (FermatNo‘𝑁)) |
| 35 | 34 | breq1d 5153 |
. . . 4
⊢ (𝑛 = 𝑁 → ((FermatNo‘𝑛) ∥ ∏𝑘 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑘) ↔ (FermatNo‘𝑁) ∥ ∏𝑘 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑘))) |
| 36 | 35 | rspcv 3618 |
. . 3
⊢ (𝑁 ∈ (0...((𝑁 + 𝑀) − 1)) → (∀𝑛 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑛) ∥ ∏𝑘 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑘) → (FermatNo‘𝑁) ∥ ∏𝑘 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑘))) |
| 37 | 19, 33, 36 | sylc 65 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (FermatNo‘𝑁)
∥ ∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘)) |
| 38 | | elfznn0 13660 |
. . . . . . 7
⊢ (𝑘 ∈ (0...((𝑁 + 𝑀) − 1)) → 𝑘 ∈ ℕ0) |
| 39 | 38 | adantl 481 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
∧ 𝑘 ∈ (0...((𝑁 + 𝑀) − 1))) → 𝑘 ∈ ℕ0) |
| 40 | | fmtnonn 47518 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
→ (FermatNo‘𝑘)
∈ ℕ) |
| 41 | 39, 40 | syl 17 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
∧ 𝑘 ∈ (0...((𝑁 + 𝑀) − 1))) → (FermatNo‘𝑘) ∈
ℕ) |
| 42 | 41 | nncnd 12282 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
∧ 𝑘 ∈ (0...((𝑁 + 𝑀) − 1))) → (FermatNo‘𝑘) ∈
ℂ) |
| 43 | 20, 42 | fprodcl 15988 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ ∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘)
∈ ℂ) |
| 44 | | 2cnd 12344 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ 2 ∈ ℂ) |
| 45 | | nn0cn 12536 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
| 46 | | nncn 12274 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℂ) |
| 47 | | addcl 11237 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + 𝑀) ∈ ℂ) |
| 48 | 45, 46, 47 | syl2an 596 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (𝑁 + 𝑀) ∈
ℂ) |
| 49 | | npcan1 11688 |
. . . . . . 7
⊢ ((𝑁 + 𝑀) ∈ ℂ → (((𝑁 + 𝑀) − 1) + 1) = (𝑁 + 𝑀)) |
| 50 | 48, 49 | syl 17 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (((𝑁 + 𝑀) − 1) + 1) = (𝑁 + 𝑀)) |
| 51 | 50 | eqcomd 2743 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (𝑁 + 𝑀) = (((𝑁 + 𝑀) − 1) + 1)) |
| 52 | 51 | fveq2d 6910 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (FermatNo‘(𝑁 +
𝑀)) =
(FermatNo‘(((𝑁 +
𝑀) − 1) +
1))) |
| 53 | | fmtnorec2 47530 |
. . . . 5
⊢ (((𝑁 + 𝑀) − 1) ∈ ℕ0
→ (FermatNo‘(((𝑁
+ 𝑀) − 1) + 1)) =
(∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘) +
2)) |
| 54 | 4, 53 | syl 17 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (FermatNo‘(((𝑁
+ 𝑀) − 1) + 1)) =
(∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘) +
2)) |
| 55 | 52, 54 | eqtrd 2777 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (FermatNo‘(𝑁 +
𝑀)) = (∏𝑘 ∈ (0...((𝑁 + 𝑀) − 1))(FermatNo‘𝑘) + 2)) |
| 56 | 43, 44, 55 | mvrraddd 11675 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ ((FermatNo‘(𝑁
+ 𝑀)) − 2) =
∏𝑘 ∈
(0...((𝑁 + 𝑀) −
1))(FermatNo‘𝑘)) |
| 57 | 37, 56 | breqtrrd 5171 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈ ℕ)
→ (FermatNo‘𝑁)
∥ ((FermatNo‘(𝑁
+ 𝑀)) −
2)) |