Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
iccpartgtprec.p |
⊢ ( 𝜑 → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
3 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 1 ) |
4 |
|
oveq2 |
⊢ ( 𝑀 = 1 → ( 1 ..^ 𝑀 ) = ( 1 ..^ 1 ) ) |
5 |
|
fzo0 |
⊢ ( 1 ..^ 1 ) = ∅ |
6 |
4 5
|
eqtrdi |
⊢ ( 𝑀 = 1 → ( 1 ..^ 𝑀 ) = ∅ ) |
7 |
|
fveq2 |
⊢ ( 𝑀 = 1 → ( 𝑃 ‘ 𝑀 ) = ( 𝑃 ‘ 1 ) ) |
8 |
7
|
breq2d |
⊢ ( 𝑀 = 1 → ( ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 1 ) ) ) |
9 |
6 8
|
raleqbidv |
⊢ ( 𝑀 = 1 → ( ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ↔ ∀ 𝑖 ∈ ∅ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 1 ) ) ) |
10 |
3 9
|
mpbiri |
⊢ ( 𝑀 = 1 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
11 |
10
|
2a1d |
⊢ ( 𝑀 = 1 → ( 𝜑 → ( 𝑀 ∈ ℕ → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) ) |
12 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ ) |
13 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
15 |
|
nnnn0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℕ0 ) |
16 |
|
nn0fz0 |
⊢ ( 𝑀 ∈ ℕ0 ↔ 𝑀 ∈ ( 0 ... 𝑀 ) ) |
17 |
15 16
|
sylib |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
19 |
12 14 18
|
iccpartxr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ) |
20 |
|
elxr |
⊢ ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ* ↔ ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∨ ( 𝑃 ‘ 𝑀 ) = +∞ ∨ ( 𝑃 ‘ 𝑀 ) = -∞ ) ) |
21 |
|
elfzoelz |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ℤ ) |
22 |
21
|
ad2antll |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑖 ∈ ℤ ) |
23 |
|
elfzo2 |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ↔ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) ) |
24 |
|
eluzelz |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → 𝑖 ∈ ℤ ) |
25 |
24
|
peano2zd |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑖 + 1 ) ∈ ℤ ) |
26 |
25
|
3ad2ant1 |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( 𝑖 + 1 ) ∈ ℤ ) |
27 |
|
simp2 |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 𝑀 ∈ ℤ ) |
28 |
|
zltp1le |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑖 < 𝑀 ↔ ( 𝑖 + 1 ) ≤ 𝑀 ) ) |
29 |
24 28
|
sylan |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑖 < 𝑀 ↔ ( 𝑖 + 1 ) ≤ 𝑀 ) ) |
30 |
29
|
biimp3a |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( 𝑖 + 1 ) ≤ 𝑀 ) |
31 |
|
eluz2 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ↔ ( ( 𝑖 + 1 ) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ( 𝑖 + 1 ) ≤ 𝑀 ) ) |
32 |
26 27 30 31
|
syl3anbrc |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → 𝑀 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ) |
33 |
23 32
|
sylbi |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑀 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ) |
34 |
33
|
ad2antll |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑀 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ) |
35 |
|
fveq2 |
⊢ ( 𝑘 = 𝑀 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 𝑀 ) ) |
36 |
35
|
eqcomd |
⊢ ( 𝑘 = 𝑀 → ( 𝑃 ‘ 𝑀 ) = ( 𝑃 ‘ 𝑘 ) ) |
37 |
36
|
eleq1d |
⊢ ( 𝑘 = 𝑀 → ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ↔ ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
38 |
37
|
biimpcd |
⊢ ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ → ( 𝑘 = 𝑀 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑘 = 𝑀 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
40 |
39
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → ( 𝑘 = 𝑀 → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
41 |
40
|
com12 |
⊢ ( 𝑘 = 𝑀 → ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
42 |
12
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑀 ∈ ℕ ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑀 ∈ ℕ ) |
44 |
43
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → 𝑀 ∈ ℕ ) |
45 |
44
|
adantl |
⊢ ( ( ¬ 𝑘 = 𝑀 ∧ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) ) → 𝑀 ∈ ℕ ) |
46 |
14
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
47 |
46
|
adantl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
48 |
47
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
49 |
48
|
adantl |
⊢ ( ( ¬ 𝑘 = 𝑀 ∧ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
50 |
|
elfz2 |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) ↔ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) ) |
51 |
|
eluz2 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ↔ ( 1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 1 ≤ 𝑖 ) ) |
52 |
|
1red |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → 1 ∈ ℝ ) |
53 |
|
zre |
⊢ ( 𝑖 ∈ ℤ → 𝑖 ∈ ℝ ) |
54 |
53
|
adantr |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → 𝑖 ∈ ℝ ) |
55 |
|
zre |
⊢ ( 𝑘 ∈ ℤ → 𝑘 ∈ ℝ ) |
56 |
55
|
adantl |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → 𝑘 ∈ ℝ ) |
57 |
|
letr |
⊢ ( ( 1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘 ) → 1 ≤ 𝑘 ) ) |
58 |
52 54 56 57
|
syl3anc |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘 ) → 1 ≤ 𝑘 ) ) |
59 |
58
|
expcomd |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑖 ≤ 𝑘 → ( 1 ≤ 𝑖 → 1 ≤ 𝑘 ) ) ) |
60 |
59
|
adantrd |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) → ( 1 ≤ 𝑖 → 1 ≤ 𝑘 ) ) ) |
61 |
60
|
3adant2 |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) → ( 1 ≤ 𝑖 → 1 ≤ 𝑘 ) ) ) |
62 |
61
|
imp |
⊢ ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → ( 1 ≤ 𝑖 → 1 ≤ 𝑘 ) ) |
63 |
62
|
com12 |
⊢ ( 1 ≤ 𝑖 → ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → 1 ≤ 𝑘 ) ) |
64 |
63
|
3ad2ant3 |
⊢ ( ( 1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 1 ≤ 𝑖 ) → ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → 1 ≤ 𝑘 ) ) |
65 |
51 64
|
sylbi |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → 1 ≤ 𝑘 ) ) |
66 |
65
|
3ad2ant1 |
⊢ ( ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀 ) → ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → 1 ≤ 𝑘 ) ) |
67 |
23 66
|
sylbi |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → 1 ≤ 𝑘 ) ) |
68 |
50 67
|
syl5bi |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → 1 ≤ 𝑘 ) ) |
69 |
68
|
imp |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → 1 ≤ 𝑘 ) |
70 |
69
|
3adant3 |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → 1 ≤ 𝑘 ) |
71 |
|
zre |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ ) |
72 |
71 55
|
anim12ci |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ) ) |
73 |
72
|
3adant1 |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ) ) |
74 |
|
ltlen |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( 𝑘 < 𝑀 ↔ ( 𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘 ) ) ) |
75 |
73 74
|
syl |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 < 𝑀 ↔ ( 𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘 ) ) ) |
76 |
|
nesym |
⊢ ( 𝑀 ≠ 𝑘 ↔ ¬ 𝑘 = 𝑀 ) |
77 |
76
|
anbi2i |
⊢ ( ( 𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘 ) ↔ ( 𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀 ) ) |
78 |
75 77
|
bitr2di |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀 ) ↔ 𝑘 < 𝑀 ) ) |
79 |
78
|
biimpd |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀 ) → 𝑘 < 𝑀 ) ) |
80 |
79
|
expd |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ≤ 𝑀 → ( ¬ 𝑘 = 𝑀 → 𝑘 < 𝑀 ) ) ) |
81 |
80
|
adantld |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) → ( ¬ 𝑘 = 𝑀 → 𝑘 < 𝑀 ) ) ) |
82 |
81
|
imp |
⊢ ( ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀 ) ) → ( ¬ 𝑘 = 𝑀 → 𝑘 < 𝑀 ) ) |
83 |
50 82
|
sylbi |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → ( ¬ 𝑘 = 𝑀 → 𝑘 < 𝑀 ) ) |
84 |
83
|
imp |
⊢ ( ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → 𝑘 < 𝑀 ) |
85 |
84
|
3adant1 |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → 𝑘 < 𝑀 ) |
86 |
70 85
|
jca |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → ( 1 ≤ 𝑘 ∧ 𝑘 < 𝑀 ) ) |
87 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → 𝑘 ∈ ℤ ) |
88 |
|
1zzd |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → 1 ∈ ℤ ) |
89 |
|
elfzel2 |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
90 |
87 88 89
|
3jca |
⊢ ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → ( 𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) |
91 |
90
|
3ad2ant2 |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → ( 𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) |
92 |
|
elfzo |
⊢ ( ( 𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑘 ∈ ( 1 ..^ 𝑀 ) ↔ ( 1 ≤ 𝑘 ∧ 𝑘 < 𝑀 ) ) ) |
93 |
91 92
|
syl |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → ( 𝑘 ∈ ( 1 ..^ 𝑀 ) ↔ ( 1 ≤ 𝑘 ∧ 𝑘 < 𝑀 ) ) ) |
94 |
86 93
|
mpbird |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ∧ ¬ 𝑘 = 𝑀 ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) |
95 |
94
|
3exp |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → ( ¬ 𝑘 = 𝑀 → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) ) ) |
96 |
95
|
ad2antll |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑘 ∈ ( 𝑖 ... 𝑀 ) → ( ¬ 𝑘 = 𝑀 → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) ) ) |
97 |
96
|
imp |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → ( ¬ 𝑘 = 𝑀 → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) ) |
98 |
97
|
impcom |
⊢ ( ( ¬ 𝑘 = 𝑀 ∧ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) |
99 |
45 49 98
|
iccpartipre |
⊢ ( ( ¬ 𝑘 = 𝑀 ∧ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) |
100 |
99
|
ex |
⊢ ( ¬ 𝑘 = 𝑀 → ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) ) |
101 |
41 100
|
pm2.61i |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ℝ ) |
102 |
43
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → 𝑀 ∈ ℕ ) |
103 |
47
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
104 |
|
1eluzge0 |
⊢ 1 ∈ ( ℤ≥ ‘ 0 ) |
105 |
|
fzoss1 |
⊢ ( 1 ∈ ( ℤ≥ ‘ 0 ) → ( 1 ..^ 𝑀 ) ⊆ ( 0 ..^ 𝑀 ) ) |
106 |
104 105
|
mp1i |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → ( 1 ..^ 𝑀 ) ⊆ ( 0 ..^ 𝑀 ) ) |
107 |
|
elfzoel2 |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑀 ∈ ℤ ) |
108 |
|
fzoval |
⊢ ( 𝑀 ∈ ℤ → ( 𝑖 ..^ 𝑀 ) = ( 𝑖 ... ( 𝑀 − 1 ) ) ) |
109 |
107 108
|
syl |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑖 ..^ 𝑀 ) = ( 𝑖 ... ( 𝑀 − 1 ) ) ) |
110 |
109
|
eqcomd |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑖 ... ( 𝑀 − 1 ) ) = ( 𝑖 ..^ 𝑀 ) ) |
111 |
110
|
eleq2d |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ↔ 𝑘 ∈ ( 𝑖 ..^ 𝑀 ) ) ) |
112 |
|
elfzouz |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ( ℤ≥ ‘ 1 ) ) |
113 |
|
fzoss1 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑖 ..^ 𝑀 ) ⊆ ( 1 ..^ 𝑀 ) ) |
114 |
112 113
|
syl |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑖 ..^ 𝑀 ) ⊆ ( 1 ..^ 𝑀 ) ) |
115 |
114
|
sseld |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ..^ 𝑀 ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) ) |
116 |
111 115
|
sylbid |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) ) |
117 |
116
|
imp |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → 𝑘 ∈ ( 1 ..^ 𝑀 ) ) |
118 |
106 117
|
sseldd |
⊢ ( ( 𝑖 ∈ ( 1 ..^ 𝑀 ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) |
119 |
118
|
ex |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → ( 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
120 |
119
|
ad2antll |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) ) |
121 |
120
|
imp |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → 𝑘 ∈ ( 0 ..^ 𝑀 ) ) |
122 |
|
iccpartimp |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑃 ∈ ( RePart ‘ 𝑀 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
123 |
102 103 121 122
|
syl3anc |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
124 |
123
|
simprd |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) ∧ 𝑘 ∈ ( 𝑖 ... ( 𝑀 − 1 ) ) ) → ( 𝑃 ‘ 𝑘 ) < ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
125 |
22 34 101 124
|
smonoord |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
126 |
125
|
ex |
⊢ ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ → ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
127 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑖 ∈ ( 1 ..^ 𝑀 ) ) |
128 |
42 46 127
|
iccpartipre |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) ∈ ℝ ) |
129 |
|
ltpnf |
⊢ ( ( 𝑃 ‘ 𝑖 ) ∈ ℝ → ( 𝑃 ‘ 𝑖 ) < +∞ ) |
130 |
128 129
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < +∞ ) |
131 |
|
breq2 |
⊢ ( ( 𝑃 ‘ 𝑀 ) = +∞ → ( ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑃 ‘ 𝑖 ) < +∞ ) ) |
132 |
130 131
|
syl5ibr |
⊢ ( ( 𝑃 ‘ 𝑀 ) = +∞ → ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
133 |
42
|
adantl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑀 ∈ ℕ ) |
134 |
46
|
adantl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑃 ∈ ( RePart ‘ 𝑀 ) ) |
135 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 1 ..^ 𝑀 ) → 𝑖 ∈ ( 1 ... 𝑀 ) ) |
136 |
135
|
ad2antll |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑖 ∈ ( 1 ... 𝑀 ) ) |
137 |
|
elfzubelfz |
⊢ ( 𝑖 ∈ ( 1 ... 𝑀 ) → 𝑀 ∈ ( 1 ... 𝑀 ) ) |
138 |
136 137
|
syl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → 𝑀 ∈ ( 1 ... 𝑀 ) ) |
139 |
133 134 138
|
iccpartgtprec |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑃 ‘ ( 𝑀 − 1 ) ) < ( 𝑃 ‘ 𝑀 ) ) |
140 |
|
breq2 |
⊢ ( -∞ = ( 𝑃 ‘ 𝑀 ) → ( ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ↔ ( 𝑃 ‘ ( 𝑀 − 1 ) ) < ( 𝑃 ‘ 𝑀 ) ) ) |
141 |
140
|
eqcoms |
⊢ ( ( 𝑃 ‘ 𝑀 ) = -∞ → ( ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ↔ ( 𝑃 ‘ ( 𝑀 − 1 ) ) < ( 𝑃 ‘ 𝑀 ) ) ) |
142 |
141
|
adantr |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ↔ ( 𝑃 ‘ ( 𝑀 − 1 ) ) < ( 𝑃 ‘ 𝑀 ) ) ) |
143 |
139 142
|
mpbird |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ) |
144 |
15
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ0 ) |
145 |
144
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 𝑀 ∈ ℕ0 ) |
146 |
|
nnne0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ≠ 0 ) |
147 |
146
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑀 ≠ 0 ) |
148 |
|
df-ne |
⊢ ( 𝑀 ≠ 1 ↔ ¬ 𝑀 = 1 ) |
149 |
148
|
biimpri |
⊢ ( ¬ 𝑀 = 1 → 𝑀 ≠ 1 ) |
150 |
149
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → 𝑀 ≠ 1 ) |
151 |
150
|
adantr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → 𝑀 ≠ 1 ) |
152 |
144 147 151
|
3jca |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → ( 𝑀 ∈ ℕ0 ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1 ) ) |
153 |
152
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑀 ∈ ℕ0 ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1 ) ) |
154 |
|
nn0n0n1ge2 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1 ) → 2 ≤ 𝑀 ) |
155 |
153 154
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → 2 ≤ 𝑀 ) |
156 |
145 155
|
jca |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑀 ∈ ℕ0 ∧ 2 ≤ 𝑀 ) ) |
157 |
156
|
adantl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑀 ∈ ℕ0 ∧ 2 ≤ 𝑀 ) ) |
158 |
|
ige2m1fz |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 2 ≤ 𝑀 ) → ( 𝑀 − 1 ) ∈ ( 0 ... 𝑀 ) ) |
159 |
157 158
|
syl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑀 − 1 ) ∈ ( 0 ... 𝑀 ) ) |
160 |
133 134 159
|
iccpartxr |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑃 ‘ ( 𝑀 − 1 ) ) ∈ ℝ* ) |
161 |
|
nltmnf |
⊢ ( ( 𝑃 ‘ ( 𝑀 − 1 ) ) ∈ ℝ* → ¬ ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ) |
162 |
160 161
|
syl |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ¬ ( 𝑃 ‘ ( 𝑀 − 1 ) ) < -∞ ) |
163 |
143 162
|
pm2.21dd |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) = -∞ ∧ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
164 |
163
|
ex |
⊢ ( ( 𝑃 ‘ 𝑀 ) = -∞ → ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
165 |
126 132 164
|
3jaoi |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∨ ( 𝑃 ‘ 𝑀 ) = +∞ ∨ ( 𝑃 ‘ 𝑀 ) = -∞ ) → ( ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
166 |
165
|
impl |
⊢ ( ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∨ ( 𝑃 ‘ 𝑀 ) = +∞ ∨ ( 𝑃 ‘ 𝑀 ) = -∞ ) ∧ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ) ∧ 𝑖 ∈ ( 1 ..^ 𝑀 ) ) → ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
167 |
166
|
ralrimiva |
⊢ ( ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∨ ( 𝑃 ‘ 𝑀 ) = +∞ ∨ ( 𝑃 ‘ 𝑀 ) = -∞ ) ∧ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
168 |
167
|
ex |
⊢ ( ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ ∨ ( 𝑃 ‘ 𝑀 ) = +∞ ∨ ( 𝑃 ‘ 𝑀 ) = -∞ ) → ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
169 |
20 168
|
sylbi |
⊢ ( ( 𝑃 ‘ 𝑀 ) ∈ ℝ* → ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
170 |
19 169
|
mpcom |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) ∧ 𝑀 ∈ ℕ ) → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |
171 |
170
|
ex |
⊢ ( ( 𝜑 ∧ ¬ 𝑀 = 1 ) → ( 𝑀 ∈ ℕ → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
172 |
171
|
expcom |
⊢ ( ¬ 𝑀 = 1 → ( 𝜑 → ( 𝑀 ∈ ℕ → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) ) |
173 |
11 172
|
pm2.61i |
⊢ ( 𝜑 → ( 𝑀 ∈ ℕ → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) ) |
174 |
1 173
|
mpd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 1 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ 𝑀 ) ) |