| Step | Hyp | Ref
| Expression |
| 1 | | ral0 4513 |
. . . 4
⊢
∀𝑖 ∈
∅ (𝑃‘0) <
(𝑃‘𝑖) |
| 2 | | oveq2 7439 |
. . . . . 6
⊢ (𝑀 = 1 → (1..^𝑀) = (1..^1)) |
| 3 | | fzo0 13723 |
. . . . . 6
⊢ (1..^1) =
∅ |
| 4 | 2, 3 | eqtrdi 2793 |
. . . . 5
⊢ (𝑀 = 1 → (1..^𝑀) = ∅) |
| 5 | 4 | raleqdv 3326 |
. . . 4
⊢ (𝑀 = 1 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝑃‘0) < (𝑃‘𝑖))) |
| 6 | 1, 5 | mpbiri 258 |
. . 3
⊢ (𝑀 = 1 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
| 7 | 6 | a1d 25 |
. 2
⊢ (𝑀 = 1 → (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 8 | | iccpartgtprec.m |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 9 | | iccpartgtprec.p |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
| 10 | 8 | nnnn0d 12587 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 11 | | 0elfz 13664 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ0
→ 0 ∈ (0...𝑀)) |
| 12 | 10, 11 | syl 17 |
. . . . . 6
⊢ (𝜑 → 0 ∈ (0...𝑀)) |
| 13 | 8, 9, 12 | iccpartxr 47406 |
. . . . 5
⊢ (𝜑 → (𝑃‘0) ∈
ℝ*) |
| 14 | 13 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈
ℝ*) |
| 15 | | elxr 13158 |
. . . . 5
⊢ ((𝑃‘0) ∈
ℝ* ↔ ((𝑃‘0) ∈ ℝ ∨ (𝑃‘0) = +∞ ∨ (𝑃‘0) =
-∞)) |
| 16 | | 0zd 12625 |
. . . . . . . . 9
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 0 ∈ ℤ) |
| 17 | | elfzouz 13703 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈
(ℤ≥‘1)) |
| 18 | | 0p1e1 12388 |
. . . . . . . . . . . 12
⊢ (0 + 1) =
1 |
| 19 | 18 | fveq2i 6909 |
. . . . . . . . . . 11
⊢
(ℤ≥‘(0 + 1)) =
(ℤ≥‘1) |
| 20 | 17, 19 | eleqtrrdi 2852 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ≥‘(0 +
1))) |
| 21 | 20 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (ℤ≥‘(0 +
1))) |
| 22 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) |
| 23 | 22 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (𝑃‘0) = (𝑃‘𝑘)) |
| 24 | 23 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑘 = 0 → ((𝑃‘0) ∈ ℝ ↔ (𝑃‘𝑘) ∈ ℝ)) |
| 25 | 24 | biimpcd 249 |
. . . . . . . . . . 11
⊢ ((𝑃‘0) ∈ ℝ →
(𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ)) |
| 26 | 25 | ad3antrrr 730 |
. . . . . . . . . 10
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ)) |
| 27 | 8 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑀 ∈ ℕ) |
| 28 | 9 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑃 ∈ (RePart‘𝑀)) |
| 29 | | elfz2nn0 13658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑖) ↔ (𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0
∧ 𝑘 ≤ 𝑖)) |
| 30 | | elfzo2 13702 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ≥‘1)
∧ 𝑀 ∈ ℤ
∧ 𝑖 < 𝑀)) |
| 31 | | simpl1 1192 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ ℕ0) |
| 32 | | simpr2 1196 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℤ) |
| 33 | | nn0ge0 12551 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑖 ∈ ℕ0
→ 0 ≤ 𝑖) |
| 34 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) → 0 ∈
ℝ) |
| 35 | | eluzelre 12889 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑖 ∈
(ℤ≥‘1) → 𝑖 ∈ ℝ) |
| 36 | 35 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑖 ∈ ℝ) |
| 37 | | zre 12617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℝ) |
| 38 | 37 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ) |
| 39 | | lelttr 11351 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((0
∈ ℝ ∧ 𝑖
∈ ℝ ∧ 𝑀
∈ ℝ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀)) |
| 40 | 34, 36, 38, 39 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀)) |
| 41 | 40 | expcomd 416 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) → (𝑖 < 𝑀 → (0 ≤ 𝑖 → 0 < 𝑀))) |
| 42 | 41 | 3impia 1118 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (0 ≤ 𝑖 → 0 < 𝑀)) |
| 43 | 33, 42 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 ∈ ℕ0
→ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀)) |
| 44 | 43 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) → ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀)) |
| 45 | 44 | imp 406 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 0 < 𝑀) |
| 46 | | elnnz 12623 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 0 <
𝑀)) |
| 47 | 32, 45, 46 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℕ) |
| 48 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
| 49 | 48 | ad2antrl 728 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0))
→ 𝑘 ∈
ℝ) |
| 50 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑖 ∈ ℕ0
→ 𝑖 ∈
ℝ) |
| 51 | 50 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0) → 𝑖 ∈ ℝ) |
| 52 | 51 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0))
→ 𝑖 ∈
ℝ) |
| 53 | 38 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0))
→ 𝑀 ∈
ℝ) |
| 54 | | lelttr 11351 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((𝑘 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 𝑘 < 𝑀)) |
| 55 | 54 | expd 415 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀))) |
| 56 | 49, 52, 53, 55 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0))
→ (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀))) |
| 57 | 56 | exp31 419 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑖 ∈
(ℤ≥‘1) → (𝑀 ∈ ℤ → ((𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0)
→ (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀))))) |
| 58 | 57 | com34 91 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑖 ∈
(ℤ≥‘1) → (𝑀 ∈ ℤ → (𝑘 ≤ 𝑖 → ((𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0)
→ (𝑖 < 𝑀 → 𝑘 < 𝑀))))) |
| 59 | 58 | com35 98 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑖 ∈
(ℤ≥‘1) → (𝑀 ∈ ℤ → (𝑖 < 𝑀 → ((𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0)
→ (𝑘 ≤ 𝑖 → 𝑘 < 𝑀))))) |
| 60 | 59 | 3imp 1111 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → ((𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0)
→ (𝑘 ≤ 𝑖 → 𝑘 < 𝑀))) |
| 61 | 60 | expdcom 414 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ ℕ0
→ (𝑖 ∈
ℕ0 → ((𝑖 ∈ (ℤ≥‘1)
∧ 𝑀 ∈ ℤ
∧ 𝑖 < 𝑀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀)))) |
| 62 | 61 | com34 91 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℕ0
→ (𝑖 ∈
ℕ0 → (𝑘 ≤ 𝑖 → ((𝑖 ∈ (ℤ≥‘1)
∧ 𝑀 ∈ ℤ
∧ 𝑖 < 𝑀) → 𝑘 < 𝑀)))) |
| 63 | 62 | 3imp1 1348 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 < 𝑀) |
| 64 | | elfzo0 13740 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0..^𝑀) ↔ (𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝑘 < 𝑀)) |
| 65 | 31, 47, 63, 64 | syl3anbrc 1344 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) ∧ (𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ (0..^𝑀)) |
| 66 | 65 | ex 412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) → ((𝑖 ∈
(ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑘 ∈ (0..^𝑀))) |
| 67 | 30, 66 | biimtrid 242 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ ℕ0
∧ 𝑖 ∈
ℕ0 ∧ 𝑘
≤ 𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀))) |
| 68 | 29, 67 | sylbi 217 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀))) |
| 69 | 68 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀))) |
| 70 | 69 | impcom 407 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (0..^𝑀)) |
| 71 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → 𝑘 ≠ 0) |
| 72 | 71 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ≠ 0) |
| 73 | | fzo1fzo0n0 13754 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (0..^𝑀) ∧ 𝑘 ≠ 0)) |
| 74 | 70, 72, 73 | sylanbrc 583 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (1..^𝑀)) |
| 75 | 74 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑘 ∈ (1..^𝑀)) |
| 76 | 27, 28, 75 | iccpartipre 47408 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → (𝑃‘𝑘) ∈ ℝ) |
| 77 | 76 | exp32 420 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ))) |
| 78 | 77 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ (((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ))) |
| 79 | 78 | imp 406 |
. . . . . . . . . . 11
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ)) |
| 80 | 79 | expdimp 452 |
. . . . . . . . . 10
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 ≠ 0 → (𝑃‘𝑘) ∈ ℝ)) |
| 81 | 26, 80 | pm2.61dne 3028 |
. . . . . . . . 9
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑃‘𝑘) ∈ ℝ) |
| 82 | 8 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ) |
| 83 | 82 | ad3antlr 731 |
. . . . . . . . . . 11
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℕ) |
| 84 | 9 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀)) |
| 85 | 84 | ad3antlr 731 |
. . . . . . . . . . 11
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑃 ∈ (RePart‘𝑀)) |
| 86 | | elfzoelz 13699 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ ℤ) |
| 87 | 86 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ ℤ) |
| 88 | | fzoval 13700 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ ℤ →
(0..^𝑖) = (0...(𝑖 − 1))) |
| 89 | 88 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ ℤ →
(0...(𝑖 − 1)) =
(0..^𝑖)) |
| 90 | 87, 89 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0...(𝑖 − 1)) = (0..^𝑖)) |
| 91 | 90 | eleq2d 2827 |
. . . . . . . . . . . . 13
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) ↔ 𝑘 ∈ (0..^𝑖))) |
| 92 | | elfzouz2 13714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ≥‘𝑖)) |
| 93 | 92 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ (ℤ≥‘𝑖)) |
| 94 | | fzoss2 13727 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘𝑖) → (0..^𝑖) ⊆ (0..^𝑀)) |
| 95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0..^𝑖) ⊆ (0..^𝑀)) |
| 96 | 95 | sseld 3982 |
. . . . . . . . . . . . 13
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0..^𝑖) → 𝑘 ∈ (0..^𝑀))) |
| 97 | 91, 96 | sylbid 240 |
. . . . . . . . . . . 12
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) → 𝑘 ∈ (0..^𝑀))) |
| 98 | 97 | imp 406 |
. . . . . . . . . . 11
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑘 ∈ (0..^𝑀)) |
| 99 | | iccpartimp 47404 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*
↑m (0...𝑀))
∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1)))) |
| 100 | 83, 85, 98, 99 | syl3anc 1373 |
. . . . . . . . . 10
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → (𝑃 ∈ (ℝ*
↑m (0...𝑀))
∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1)))) |
| 101 | 100 | simprd 495 |
. . . . . . . . 9
⊢
(((((𝑃‘0)
∈ ℝ ∧ (𝜑 ∧
¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))) |
| 102 | 16, 21, 81, 101 | smonoord 47358 |
. . . . . . . 8
⊢ ((((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖)) |
| 103 | 102 | ralrimiva 3146 |
. . . . . . 7
⊢ (((𝑃‘0) ∈ ℝ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
| 104 | 103 | ex 412 |
. . . . . 6
⊢ ((𝑃‘0) ∈ ℝ →
((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 105 | | lbfzo0 13739 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
(0..^𝑀) ↔ 𝑀 ∈
ℕ) |
| 106 | 8, 105 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 0 ∈ (0..^𝑀)) |
| 107 | 8, 9, 106 | 3jca 1129 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀))) |
| 108 | 107 | ad2antrl 728 |
. . . . . . . . . . . . 13
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀))) |
| 109 | 108 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀))) |
| 110 | | iccpartimp 47404 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*
↑m (0...𝑀))
∧ (𝑃‘0) <
(𝑃‘(0 +
1)))) |
| 111 | 109, 110 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃 ∈ (ℝ*
↑m (0...𝑀))
∧ (𝑃‘0) <
(𝑃‘(0 +
1)))) |
| 112 | 111 | simprd 495 |
. . . . . . . . . 10
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘(0 + 1))) |
| 113 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ ((𝑃‘0) = +∞ →
((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞
< (𝑃‘(0 +
1)))) |
| 114 | 113 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞
< (𝑃‘(0 +
1)))) |
| 115 | 114 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 +
1)))) |
| 116 | 112, 115 | mpbid 232 |
. . . . . . . . 9
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → +∞ < (𝑃‘(0 + 1))) |
| 117 | 8 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → 𝑀 ∈ ℕ) |
| 118 | 117 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ) |
| 119 | 9 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → 𝑃 ∈ (RePart‘𝑀)) |
| 120 | 119 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀)) |
| 121 | | 1nn0 12542 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℕ0 |
| 122 | 121 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ → 1 ∈
ℕ0) |
| 123 | | nnnn0 12533 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℕ0) |
| 124 | | nnge1 12294 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ → 1 ≤
𝑀) |
| 125 | 122, 123,
124 | 3jca 1129 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℕ → (1 ∈
ℕ0 ∧ 𝑀
∈ ℕ0 ∧ 1 ≤ 𝑀)) |
| 126 | 8, 125 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1 ∈
ℕ0 ∧ 𝑀
∈ ℕ0 ∧ 1 ≤ 𝑀)) |
| 127 | | elfz2nn0 13658 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(0...𝑀) ↔ (1 ∈
ℕ0 ∧ 𝑀
∈ ℕ0 ∧ 1 ≤ 𝑀)) |
| 128 | 126, 127 | sylibr 234 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ∈ (0...𝑀)) |
| 129 | 18, 128 | eqeltrid 2845 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0 + 1) ∈ (0...𝑀)) |
| 130 | 129 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → (0 + 1) ∈
(0...𝑀)) |
| 131 | 130 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0 + 1) ∈ (0...𝑀)) |
| 132 | 118, 120,
131 | iccpartxr 47406 |
. . . . . . . . . 10
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘(0 + 1)) ∈
ℝ*) |
| 133 | | pnfnlt 13170 |
. . . . . . . . . 10
⊢ ((𝑃‘(0 + 1)) ∈
ℝ* → ¬ +∞ < (𝑃‘(0 + 1))) |
| 134 | 132, 133 | syl 17 |
. . . . . . . . 9
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ¬ +∞ < (𝑃‘(0 +
1))) |
| 135 | 116, 134 | pm2.21dd 195 |
. . . . . . . 8
⊢ ((((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖)) |
| 136 | 135 | ralrimiva 3146 |
. . . . . . 7
⊢ (((𝑃‘0) = +∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
| 137 | 136 | ex 412 |
. . . . . 6
⊢ ((𝑃‘0) = +∞ →
((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 138 | 8 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ) |
| 139 | 9 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀)) |
| 140 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (1..^𝑀)) |
| 141 | 138, 139,
140 | iccpartipre 47408 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) ∈ ℝ) |
| 142 | | mnflt 13165 |
. . . . . . . . . . 11
⊢ ((𝑃‘𝑖) ∈ ℝ → -∞ < (𝑃‘𝑖)) |
| 143 | 141, 142 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → -∞ < (𝑃‘𝑖)) |
| 144 | 143 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖)) |
| 145 | 144 | ad2antrl 728 |
. . . . . . . 8
⊢ (((𝑃‘0) = -∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖)) |
| 146 | | breq1 5146 |
. . . . . . . . . 10
⊢ ((𝑃‘0) = -∞ →
((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖))) |
| 147 | 146 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑃‘0) = -∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖))) |
| 148 | 147 | ralbidv 3178 |
. . . . . . . 8
⊢ (((𝑃‘0) = -∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖))) |
| 149 | 145, 148 | mpbird 257 |
. . . . . . 7
⊢ (((𝑃‘0) = -∞ ∧
(𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
| 150 | 149 | ex 412 |
. . . . . 6
⊢ ((𝑃‘0) = -∞ →
((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 151 | 104, 137,
150 | 3jaoi 1430 |
. . . . 5
⊢ (((𝑃‘0) ∈ ℝ ∨
(𝑃‘0) = +∞ ∨
(𝑃‘0) = -∞)
→ ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 152 | 15, 151 | sylbi 217 |
. . . 4
⊢ ((𝑃‘0) ∈
ℝ* → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 153 | 14, 152 | mpcom 38 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |
| 154 | 153 | expcom 413 |
. 2
⊢ (¬
𝑀 = 1 → (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))) |
| 155 | 7, 154 | pm2.61i 182 |
1
⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) |