Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
iccpartgtprec.p |
⊢ ( 𝜑 → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
3 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) |
4 |
|
oveq2 |
⊢ ( 𝑀 = 1 → ( 1 ..^ 𝑀 ) = ( 1 ..^ 1 ) ) |
5 |
|
fzo0 |
⊢ ( 1 ..^ 1 ) = ∅ |
6 |
4 5
|
eqtrdi |
⊢ ( 𝑀 = 1 → ( 1 ..^ 𝑀 ) = ∅ ) |
7 |
6
|
raleqdv |
⊢ ( 𝑀 = 1 → ( ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ ∅ ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
8 |
3 7
|
mpbiri |
⊢ ( 𝑀 = 1 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
9 |
8
|
a1d |
⊢ ( 𝑀 = 1 → ( 𝜑 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
10 |
1
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
11 |
|
0elfz |
⊢ ( 𝑀 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑀 ) ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
13 |
1 2 12
|
iccpartxr |
⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) ∈ ℝ* ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ( 𝑃 ‘ 0 ) ∈ ℝ* ) |
15 |
|
elxr |
⊢ ( ( 𝑃 ‘ 0 ) ∈ ℝ* ↔ ( ( 𝑃 ‘ 0 ) ∈ ℝ ∨ ( 𝑃 ‘ 0 ) = +∞ ∨ ( 𝑃 ‘ 0 ) = -∞ ) ) |
16 |
|
0zd |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 0 ∈ ℤ ) |
17 |
|
elfzouz |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ( ℤ≥ ‘ 1 ) ) |
18 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
19 |
18
|
fveq2i |
⊢ ( ℤ≥ ‘ ( 0 + 1 ) ) = ( ℤ≥ ‘ 1 ) |
20 |
17 19
|
eleqtrrdi |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
21 |
20
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑖 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) ) |
23 |
22
|
eqcomd |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑘 ) ) |
24 |
23
|
eleq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 0 ) ∈ ℝ ↔ ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
25 |
24
|
biimpcd |
⊢ ( ( 𝑃 ‘ 0 ) ∈ ℝ → ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
26 |
25
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... 𝑖 ) ) → ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
27 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) ) → 𝑀 ∈ ℕ ) |
28 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
29 |
|
elfz2nn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑖 ) ↔ ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ) |
30 |
|
elfzo2 |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ↔ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) |
31 |
|
simpl1 |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
32 |
|
simpr2 |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 𝑀 ∈ ℤ ) |
33 |
|
nn0ge0 |
⊢ ( 𝑖 ∈ ℕ0 → 0 ≤ 𝑖 ) |
34 |
|
0red |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → 0 ∈ ℝ ) |
35 |
|
eluzelre |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → 𝑖 ∈ ℝ ) |
36 |
35
|
adantr |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → 𝑖 ∈ ℝ ) |
37 |
|
zre |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ ) |
38 |
37
|
adantl |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → 𝑀 ∈ ℝ ) |
39 |
|
lelttr |
⊢ ( ( 0 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( ( 0 ≤ 𝑖 ∧ 𝑖 < 𝑀 ) → 0 < 𝑀 ) ) |
40 |
34 36 38 39
|
syl3anc |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → ( ( 0 ≤ 𝑖 ∧ 𝑖 < 𝑀 ) → 0 < 𝑀 ) ) |
41 |
40
|
expcomd |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑖 < 𝑀 → ( 0 ≤ 𝑖 → 0 < 𝑀 ) ) ) |
42 |
41
|
3impia |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( 0 ≤ 𝑖 → 0 < 𝑀 ) ) |
43 |
33 42
|
syl5com |
⊢ ( 𝑖 ∈ ℕ0 → ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 0 < 𝑀 ) ) |
44 |
43
|
3ad2ant2 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) → ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 0 < 𝑀 ) ) |
45 |
44
|
imp |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 0 < 𝑀 ) |
46 |
|
elnnz |
⊢ ( 𝑀 ∈ ℕ ↔ ( 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
47 |
32 45 46
|
sylanbrc |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 𝑀 ∈ ℕ ) |
48 |
|
nn0re |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ ) |
49 |
48
|
ad2antrl |
⊢ ( ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) ) → 𝑘 ∈ ℝ ) |
50 |
|
nn0re |
⊢ ( 𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ ) |
51 |
50
|
adantl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → 𝑖 ∈ ℝ ) |
52 |
51
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) ) → 𝑖 ∈ ℝ ) |
53 |
38
|
adantr |
⊢ ( ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) ) → 𝑀 ∈ ℝ ) |
54 |
|
lelttr |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( ( 𝑘 ≤ 𝑖 ∧ 𝑖 < 𝑀 ) → 𝑘 < 𝑀 ) ) |
55 |
54
|
expd |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( 𝑘 ≤ 𝑖 → ( 𝑖 < 𝑀 → 𝑘 < 𝑀 ) ) ) |
56 |
49 52 53 55
|
syl3anc |
⊢ ( ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) ) → ( 𝑘 ≤ 𝑖 → ( 𝑖 < 𝑀 → 𝑘 < 𝑀 ) ) ) |
57 |
56
|
exp31 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑀 ∈ ℤ → ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑘 ≤ 𝑖 → ( 𝑖 < 𝑀 → 𝑘 < 𝑀 ) ) ) ) ) |
58 |
57
|
com34 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑀 ∈ ℤ → ( 𝑘 ≤ 𝑖 → ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑖 < 𝑀 → 𝑘 < 𝑀 ) ) ) ) ) |
59 |
58
|
com35 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑀 ∈ ℤ → ( 𝑖 < 𝑀 → ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑘 ≤ 𝑖 → 𝑘 < 𝑀 ) ) ) ) ) |
60 |
59
|
3imp |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑘 ≤ 𝑖 → 𝑘 < 𝑀 ) ) ) |
61 |
60
|
expdcom |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑖 ∈ ℕ0 → ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( 𝑘 ≤ 𝑖 → 𝑘 < 𝑀 ) ) ) ) |
62 |
61
|
com34 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑖 ∈ ℕ0 → ( 𝑘 ≤ 𝑖 → ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 𝑘 < 𝑀 ) ) ) ) |
63 |
62
|
3imp1 |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 𝑘 < 𝑀 ) |
64 |
|
elfzo0 |
⊢ ( 𝑘 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑘 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝑘 < 𝑀 ) ) |
65 |
31 47 63 64
|
syl3anbrc |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) ∧ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) |
66 |
65
|
ex |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) → ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
67 |
30 66
|
syl5bi |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ∧ 𝑘 ≤ 𝑖 ) → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
68 |
29 67
|
sylbi |
⊢ ( 𝑘 ∈ ( 0 ... 𝑖 ) → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
70 |
69
|
impcom |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) |
71 |
|
simpr |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) → 𝑘 ≠ 0 ) |
72 |
71
|
adantl |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) → 𝑘 ≠ 0 ) |
73 |
|
fzo1fzo0n0 |
⊢ ( 𝑘 ∈ ( 1 ..^ 𝑀 ) ↔ ( 𝑘 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ≠ 0 ) ) |
74 |
70 72 73
|
sylanbrc |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) |
75 |
74
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) |
76 |
27 28 75
|
iccpartipre |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) |
77 |
76
|
exp32 |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) ) |
78 |
77
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) ) |
79 |
78
|
imp |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( ( 𝑘 ∈ ( 0 ... 𝑖 ) ∧ 𝑘 ≠ 0 ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
80 |
79
|
expdimp |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... 𝑖 ) ) → ( 𝑘 ≠ 0 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
81 |
26 80
|
pm2.61dne |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... 𝑖 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) |
82 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → 𝑀 ∈ ℕ ) |
83 |
82
|
ad3antlr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) → 𝑀 ∈ ℕ ) |
84 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
85 |
84
|
ad3antlr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
86 |
|
elfzoelz |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ℤ ) |
87 |
86
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑖 ∈ ℤ ) |
88 |
|
fzoval |
⊢ ( 𝑖 ∈ ℤ → ( 0 ..^ 𝑖 ) = ( 0 ... ( 𝑖 − 1 ) ) ) |
89 |
88
|
eqcomd |
⊢ ( 𝑖 ∈ ℤ → ( 0 ... ( 𝑖 − 1 ) ) = ( 0 ..^ 𝑖 ) ) |
90 |
87 89
|
syl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 0 ... ( 𝑖 − 1 ) ) = ( 0 ..^ 𝑖 ) ) |
91 |
90
|
eleq2d |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) ↔ 𝑘 ∈ ( 0 ..^ 𝑖 ) ) ) |
92 |
|
elfzouz2 |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑖 ) ) |
93 |
92
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑖 ) ) |
94 |
|
fzoss2 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑖 ) → ( 0 ..^ 𝑖 ) ⊆ ( 0 ..^ 𝑀 ) ) |
95 |
93 94
|
syl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 0 ..^ 𝑖 ) ⊆ ( 0 ..^ 𝑀 ) ) |
96 |
95
|
sseld |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑘 ∈ ( 0 ..^ 𝑖 ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
97 |
91 96
|
sylbid |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
98 |
97
|
imp |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) |
99 |
|
iccpartimp |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
100 |
83 85 98 99
|
syl3anc |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ∧ 𝑘 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) → ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
102 |
16 21 81 101
|
smonoord |
⊢ ( ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
103 |
102
|
ralrimiva |
⊢ ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
104 |
103
|
ex |
⊢ ( ( 𝑃 ‘ 0 ) ∈ ℝ → ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
105 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ 𝑀 ∈ ℕ ) |
106 |
1 105
|
sylibr |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
107 |
1 2 106
|
3jca |
⊢ ( 𝜑 → ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) ) |
108 |
107
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) ) |
109 |
108
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) ) |
110 |
|
iccpartimp |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ) ) |
111 |
109 110
|
syl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ) ) |
112 |
111
|
simprd |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ) |
113 |
|
breq1 |
⊢ ( ( 𝑃 ‘ 0 ) = +∞ → ( ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ↔ +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) ) |
114 |
113
|
adantr |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ↔ +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) ) |
115 |
114
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ ( 0 + 1 ) ) ↔ +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) ) |
116 |
112 115
|
mpbid |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) |
117 |
1
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → 𝑀 ∈ ℕ ) |
118 |
117
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑀 ∈ ℕ ) |
119 |
2
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
120 |
119
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
121 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
122 |
121
|
a1i |
⊢ ( 𝑀 ∈ ℕ → 1 ∈ ℕ0 ) |
123 |
|
nnnn0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℕ0 ) |
124 |
|
nnge1 |
⊢ ( 𝑀 ∈ ℕ → 1 ≤ 𝑀 ) |
125 |
122 123 124
|
3jca |
⊢ ( 𝑀 ∈ ℕ → ( 1 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 1 ≤ 𝑀 ) ) |
126 |
1 125
|
syl |
⊢ ( 𝜑 → ( 1 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 1 ≤ 𝑀 ) ) |
127 |
|
elfz2nn0 |
⊢ ( 1 ∈ ( 0 ... 𝑀 ) ↔ ( 1 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 1 ≤ 𝑀 ) ) |
128 |
126 127
|
sylibr |
⊢ ( 𝜑 → 1 ∈ ( 0 ... 𝑀 ) ) |
129 |
18 128
|
eqeltrid |
⊢ ( 𝜑 → ( 0 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
130 |
129
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( 0 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
131 |
130
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 0 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
132 |
118 120 131
|
iccpartxr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ ( 0 + 1 ) ) ∈ ℝ* ) |
133 |
|
pnfnlt |
⊢ ( ( 𝑃 ‘ ( 0 + 1 ) ) ∈ ℝ* → ¬ +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) |
134 |
132 133
|
syl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ¬ +∞ < ( 𝑃 ‘ ( 0 + 1 ) ) ) |
135 |
116 134
|
pm2.21dd |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
136 |
135
|
ralrimiva |
⊢ ( ( ( 𝑃 ‘ 0 ) = +∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
137 |
136
|
ex |
⊢ ( ( 𝑃 ‘ 0 ) = +∞ → ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
138 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑀 ∈ ℕ ) |
139 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
140 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑖 ∈ ( 1 ..^ 𝑀 ) ) |
141 |
138 139 140
|
iccpartipre |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) ∈ ℝ ) |
142 |
|
mnflt |
⊢ ( ( 𝑃 ‘ 𝑖 ) ∈ ℝ → -∞ < ( 𝑃 ‘ 𝑖 ) ) |
143 |
141 142
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → -∞ < ( 𝑃 ‘ 𝑖 ) ) |
144 |
143
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) -∞ < ( 𝑃 ‘ 𝑖 ) ) |
145 |
144
|
ad2antrl |
⊢ ( ( ( 𝑃 ‘ 0 ) = -∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) -∞ < ( 𝑃 ‘ 𝑖 ) ) |
146 |
|
breq1 |
⊢ ( ( 𝑃 ‘ 0 ) = -∞ → ( ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ↔ -∞ < ( 𝑃 ‘ 𝑖 ) ) ) |
147 |
146
|
adantr |
⊢ ( ( ( 𝑃 ‘ 0 ) = -∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ↔ -∞ < ( 𝑃 ‘ 𝑖 ) ) ) |
148 |
147
|
ralbidv |
⊢ ( ( ( 𝑃 ‘ 0 ) = -∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ( ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) -∞ < ( 𝑃 ‘ 𝑖 ) ) ) |
149 |
145 148
|
mpbird |
⊢ ( ( ( 𝑃 ‘ 0 ) = -∞ ∧ ( 𝜑 ∧ ¬ 𝑀 = 1 ) ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
150 |
149
|
ex |
⊢ ( ( 𝑃 ‘ 0 ) = -∞ → ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
151 |
104 137 150
|
3jaoi |
⊢ ( ( ( 𝑃 ‘ 0 ) ∈ ℝ ∨ ( 𝑃 ‘ 0 ) = +∞ ∨ ( 𝑃 ‘ 0 ) = -∞ ) → ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
152 |
15 151
|
sylbi |
⊢ ( ( 𝑃 ‘ 0 ) ∈ ℝ* → ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
153 |
14 152
|
mpcom |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |
154 |
153
|
expcom |
⊢ ( ¬ 𝑀 = 1 → ( 𝜑 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) ) |
155 |
9 154
|
pm2.61i |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 0 ) < ( 𝑃 ‘ 𝑖 ) ) |