Step | Hyp | Ref
| Expression |
1 | | fveq2 6774 |
. . . . . 6
⊢ ((𝐼 + 1) = 𝐽 → (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽)) |
2 | 1 | olcd 871 |
. . . . 5
⊢ ((𝐼 + 1) = 𝐽 → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽))) |
3 | 2 | a1d 25 |
. . . 4
⊢ ((𝐼 + 1) = 𝐽 → (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽)))) |
4 | | elfzoelz 13387 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ ℤ) |
5 | | elfzoelz 13387 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ (0..^𝑀) → 𝐽 ∈ ℤ) |
6 | | zltp1le 12370 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) |
7 | 6 | biimpcd 248 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 < 𝐽 → ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 + 1) ≤ 𝐽)) |
8 | 7 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 + 1) ≤ 𝐽)) |
9 | 8 | impcom 408 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → (𝐼 + 1) ≤ 𝐽) |
10 | | df-ne 2944 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 + 1) ≠ 𝐽 ↔ ¬ (𝐼 + 1) = 𝐽) |
11 | | necom 2997 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 + 1) ≠ 𝐽 ↔ 𝐽 ≠ (𝐼 + 1)) |
12 | 10, 11 | sylbb1 236 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(𝐼 + 1) = 𝐽 → 𝐽 ≠ (𝐼 + 1)) |
13 | 12 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → 𝐽 ≠ (𝐼 + 1)) |
14 | 13 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → 𝐽 ≠ (𝐼 + 1)) |
15 | 9, 14 | jca 512 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → ((𝐼 + 1) ≤ 𝐽 ∧ 𝐽 ≠ (𝐼 + 1))) |
16 | | peano2z 12361 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐼 ∈ ℤ → (𝐼 + 1) ∈
ℤ) |
17 | 16 | zred 12426 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 ∈ ℤ → (𝐼 + 1) ∈
ℝ) |
18 | | zre 12323 |
. . . . . . . . . . . . . . . 16
⊢ (𝐽 ∈ ℤ → 𝐽 ∈
ℝ) |
19 | 17, 18 | anim12i 613 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → ((𝐼 + 1) ∈ ℝ ∧ 𝐽 ∈
ℝ)) |
20 | 19 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → ((𝐼 + 1) ∈ ℝ ∧ 𝐽 ∈ ℝ)) |
21 | | ltlen 11076 |
. . . . . . . . . . . . . 14
⊢ (((𝐼 + 1) ∈ ℝ ∧ 𝐽 ∈ ℝ) → ((𝐼 + 1) < 𝐽 ↔ ((𝐼 + 1) ≤ 𝐽 ∧ 𝐽 ≠ (𝐼 + 1)))) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → ((𝐼 + 1) < 𝐽 ↔ ((𝐼 + 1) ≤ 𝐽 ∧ 𝐽 ≠ (𝐼 + 1)))) |
23 | 15, 22 | mpbird 256 |
. . . . . . . . . . . 12
⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽)) → (𝐼 + 1) < 𝐽) |
24 | 23 | ex 413 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → (𝐼 + 1) < 𝐽)) |
25 | 4, 5, 24 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → (𝐼 + 1) < 𝐽)) |
26 | 25 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → (𝐼 + 1) < 𝐽)) |
27 | | iccpartiun.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
28 | | iccpartiun.p |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
29 | 27, 28 | iccpartgt 44879 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))) |
30 | | fzofzp1 13484 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) |
31 | | elfzofz 13403 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ (0..^𝑀) → 𝐽 ∈ (0...𝑀)) |
32 | | breq1 5077 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝐼 + 1) → (𝑖 < 𝑗 ↔ (𝐼 + 1) < 𝑗)) |
33 | | fveq2 6774 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (𝐼 + 1) → (𝑃‘𝑖) = (𝑃‘(𝐼 + 1))) |
34 | 33 | breq1d 5084 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝐼 + 1) → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘(𝐼 + 1)) < (𝑃‘𝑗))) |
35 | 32, 34 | imbi12d 345 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝐼 + 1) → ((𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) ↔ ((𝐼 + 1) < 𝑗 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝑗)))) |
36 | | breq2 5078 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐽 → ((𝐼 + 1) < 𝑗 ↔ (𝐼 + 1) < 𝐽)) |
37 | | fveq2 6774 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐽 → (𝑃‘𝑗) = (𝑃‘𝐽)) |
38 | 37 | breq2d 5086 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐽 → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝑗) ↔ (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽))) |
39 | 36, 38 | imbi12d 345 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐽 → (((𝐼 + 1) < 𝑗 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝑗)) ↔ ((𝐼 + 1) < 𝐽 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽)))) |
40 | 35, 39 | rspc2v 3570 |
. . . . . . . . . . 11
⊢ (((𝐼 + 1) ∈ (0...𝑀) ∧ 𝐽 ∈ (0...𝑀)) → (∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) → ((𝐼 + 1) < 𝐽 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽)))) |
41 | 30, 31, 40 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) → ((𝐼 + 1) < 𝐽 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽)))) |
42 | 29, 41 | mpan9 507 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → ((𝐼 + 1) < 𝐽 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽))) |
43 | 26, 42 | syld 47 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → ((𝐼 < 𝐽 ∧ ¬ (𝐼 + 1) = 𝐽) → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽))) |
44 | 43 | expdimp 453 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → (¬ (𝐼 + 1) = 𝐽 → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽))) |
45 | 44 | impcom 408 |
. . . . . 6
⊢ ((¬
(𝐼 + 1) = 𝐽 ∧ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽)) → (𝑃‘(𝐼 + 1)) < (𝑃‘𝐽)) |
46 | 45 | orcd 870 |
. . . . 5
⊢ ((¬
(𝐼 + 1) = 𝐽 ∧ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽)) → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽))) |
47 | 46 | ex 413 |
. . . 4
⊢ (¬
(𝐼 + 1) = 𝐽 → (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽)))) |
48 | 3, 47 | pm2.61i 182 |
. . 3
⊢ (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽))) |
49 | 27 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → 𝑀 ∈ ℕ) |
50 | 28 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → 𝑃 ∈ (RePart‘𝑀)) |
51 | 30 | adantr 481 |
. . . . . . . 8
⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 + 1) ∈ (0...𝑀)) |
52 | 51 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → (𝐼 + 1) ∈ (0...𝑀)) |
53 | 49, 50, 52 | iccpartxr 44871 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → (𝑃‘(𝐼 + 1)) ∈
ℝ*) |
54 | 31 | adantl 482 |
. . . . . . . 8
⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → 𝐽 ∈ (0...𝑀)) |
55 | 54 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → 𝐽 ∈ (0...𝑀)) |
56 | 49, 50, 55 | iccpartxr 44871 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → (𝑃‘𝐽) ∈
ℝ*) |
57 | 53, 56 | jca 512 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) → ((𝑃‘(𝐼 + 1)) ∈ ℝ* ∧
(𝑃‘𝐽) ∈
ℝ*)) |
58 | 57 | adantr 481 |
. . . 4
⊢ (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → ((𝑃‘(𝐼 + 1)) ∈ ℝ* ∧
(𝑃‘𝐽) ∈
ℝ*)) |
59 | | xrleloe 12878 |
. . . 4
⊢ (((𝑃‘(𝐼 + 1)) ∈ ℝ* ∧
(𝑃‘𝐽) ∈ ℝ*) → ((𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽) ↔ ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽)))) |
60 | 58, 59 | syl 17 |
. . 3
⊢ (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → ((𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽) ↔ ((𝑃‘(𝐼 + 1)) < (𝑃‘𝐽) ∨ (𝑃‘(𝐼 + 1)) = (𝑃‘𝐽)))) |
61 | 48, 60 | mpbird 256 |
. 2
⊢ (((𝜑 ∧ (𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀))) ∧ 𝐼 < 𝐽) → (𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽)) |
62 | 61 | exp31 420 |
1
⊢ (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 < 𝐽 → (𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽)))) |