Step |
Hyp |
Ref |
Expression |
1 |
|
monoords.fk |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
2 |
|
monoords.flt |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) < ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
3 |
|
monoords.i |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) |
4 |
|
monoords.j |
⊢ ( 𝜑 → 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) |
5 |
|
monoords.iltj |
⊢ ( 𝜑 → 𝐼 < 𝐽 ) |
6 |
3
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) ) |
7 |
|
eleq1 |
⊢ ( 𝑘 = 𝐼 → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) ) |
8 |
7
|
anbi2d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑘 = 𝐼 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝐼 ) ) |
10 |
9
|
eleq1d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ 𝐼 ) ∈ ℝ ) ) |
11 |
8 10
|
imbi12d |
⊢ ( 𝑘 = 𝐼 → ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝐼 ) ∈ ℝ ) ) ) |
12 |
11 1
|
vtoclg |
⊢ ( 𝐼 ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝐼 ) ∈ ℝ ) ) |
13 |
3 6 12
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐼 ) ∈ ℝ ) |
14 |
|
elfzel1 |
⊢ ( 𝐼 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ∈ ℤ ) |
15 |
3 14
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
16 |
3
|
elfzelzd |
⊢ ( 𝜑 → 𝐼 ∈ ℤ ) |
17 |
|
elfzle1 |
⊢ ( 𝐼 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ≤ 𝐼 ) |
18 |
3 17
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ 𝐼 ) |
19 |
|
eluz2 |
⊢ ( 𝐼 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼 ) ) |
20 |
15 16 18 19
|
syl3anbrc |
⊢ ( 𝜑 → 𝐼 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
21 |
|
elfzuz2 |
⊢ ( 𝐼 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
22 |
3 21
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
23 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
25 |
16
|
zred |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
26 |
4
|
elfzelzd |
⊢ ( 𝜑 → 𝐽 ∈ ℤ ) |
27 |
26
|
zred |
⊢ ( 𝜑 → 𝐽 ∈ ℝ ) |
28 |
24
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
29 |
|
elfzle2 |
⊢ ( 𝐽 ∈ ( 𝑀 ... 𝑁 ) → 𝐽 ≤ 𝑁 ) |
30 |
4 29
|
syl |
⊢ ( 𝜑 → 𝐽 ≤ 𝑁 ) |
31 |
25 27 28 5 30
|
ltletrd |
⊢ ( 𝜑 → 𝐼 < 𝑁 ) |
32 |
|
elfzo2 |
⊢ ( 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝐼 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ∧ 𝐼 < 𝑁 ) ) |
33 |
20 24 31 32
|
syl3anbrc |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) |
34 |
|
fzofzp1 |
⊢ ( 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
35 |
33 34
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
36 |
35
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
37 |
|
eleq1 |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
38 |
37
|
anbi2d |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
39 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) |
40 |
39
|
eleq1d |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ) |
41 |
38 40
|
imbi12d |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ) ) |
42 |
41 1
|
vtoclg |
⊢ ( ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝜑 ∧ ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ) |
43 |
35 36 42
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
44 |
4
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) ) |
45 |
|
eleq1 |
⊢ ( 𝑘 = 𝐽 → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) ) |
46 |
45
|
anbi2d |
⊢ ( 𝑘 = 𝐽 → ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
47 |
|
fveq2 |
⊢ ( 𝑘 = 𝐽 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝐽 ) ) |
48 |
47
|
eleq1d |
⊢ ( 𝑘 = 𝐽 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ 𝐽 ) ∈ ℝ ) ) |
49 |
46 48
|
imbi12d |
⊢ ( 𝑘 = 𝐽 → ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝐽 ) ∈ ℝ ) ) ) |
50 |
49 1
|
vtoclg |
⊢ ( 𝐽 ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝜑 ∧ 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝐽 ) ∈ ℝ ) ) |
51 |
4 44 50
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐽 ) ∈ ℝ ) |
52 |
33
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) ) |
53 |
|
eleq1 |
⊢ ( 𝑘 = 𝐼 → ( 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) ↔ 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) ) |
54 |
53
|
anbi2d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) ) ↔ ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) ) ) |
55 |
|
fvoveq1 |
⊢ ( 𝑘 = 𝐼 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) |
56 |
9 55
|
breq12d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝐹 ‘ 𝑘 ) < ( 𝐹 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐹 ‘ 𝐼 ) < ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) ) |
57 |
54 56
|
imbi12d |
⊢ ( 𝑘 = 𝐼 → ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) < ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝐼 ) < ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) ) ) |
58 |
57 2
|
vtoclg |
⊢ ( 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) → ( ( 𝜑 ∧ 𝐼 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝐼 ) < ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) ) |
59 |
33 52 58
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐼 ) < ( 𝐹 ‘ ( 𝐼 + 1 ) ) ) |
60 |
16
|
peano2zd |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ℤ ) |
61 |
|
zltp1le |
⊢ ( ( 𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ ) → ( 𝐼 < 𝐽 ↔ ( 𝐼 + 1 ) ≤ 𝐽 ) ) |
62 |
16 26 61
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 < 𝐽 ↔ ( 𝐼 + 1 ) ≤ 𝐽 ) ) |
63 |
5 62
|
mpbid |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ≤ 𝐽 ) |
64 |
|
eluz2 |
⊢ ( 𝐽 ∈ ( ℤ≥ ‘ ( 𝐼 + 1 ) ) ↔ ( ( 𝐼 + 1 ) ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ ( 𝐼 + 1 ) ≤ 𝐽 ) ) |
65 |
60 26 63 64
|
syl3anbrc |
⊢ ( 𝜑 → 𝐽 ∈ ( ℤ≥ ‘ ( 𝐼 + 1 ) ) ) |
66 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 ∈ ℤ ) |
67 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑁 ∈ ℤ ) |
68 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) → 𝑘 ∈ ℤ ) |
69 |
68
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑘 ∈ ℤ ) |
70 |
66
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 ∈ ℝ ) |
71 |
69
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑘 ∈ ℝ ) |
72 |
60
|
zred |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ℝ ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → ( 𝐼 + 1 ) ∈ ℝ ) |
74 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝐼 ∈ ℝ ) |
75 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 ≤ 𝐼 ) |
76 |
74
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝐼 < ( 𝐼 + 1 ) ) |
77 |
70 74 73 75 76
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 < ( 𝐼 + 1 ) ) |
78 |
|
elfzle1 |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) → ( 𝐼 + 1 ) ≤ 𝑘 ) |
79 |
78
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → ( 𝐼 + 1 ) ≤ 𝑘 ) |
80 |
70 73 71 77 79
|
ltletrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 < 𝑘 ) |
81 |
70 71 80
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑀 ≤ 𝑘 ) |
82 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝐽 ∈ ℝ ) |
83 |
67
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑁 ∈ ℝ ) |
84 |
|
elfzle2 |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) → 𝑘 ≤ 𝐽 ) |
85 |
84
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑘 ≤ 𝐽 ) |
86 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝐽 ≤ 𝑁 ) |
87 |
71 82 83 85 86
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑘 ≤ 𝑁 ) |
88 |
66 67 69 81 87
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
89 |
88 1
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... 𝐽 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
90 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑀 ∈ ℤ ) |
91 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑁 ∈ ℤ ) |
92 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) → 𝑘 ∈ ℤ ) |
93 |
92
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ∈ ℤ ) |
94 |
90
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑀 ∈ ℝ ) |
95 |
93
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ∈ ℝ ) |
96 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐼 + 1 ) ∈ ℝ ) |
97 |
15
|
zred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
98 |
25
|
ltp1d |
⊢ ( 𝜑 → 𝐼 < ( 𝐼 + 1 ) ) |
99 |
97 25 72 18 98
|
lelttrd |
⊢ ( 𝜑 → 𝑀 < ( 𝐼 + 1 ) ) |
100 |
99
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑀 < ( 𝐼 + 1 ) ) |
101 |
|
elfzle1 |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) → ( 𝐼 + 1 ) ≤ 𝑘 ) |
102 |
101
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐼 + 1 ) ≤ 𝑘 ) |
103 |
94 96 95 100 102
|
ltletrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑀 < 𝑘 ) |
104 |
94 95 103
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑀 ≤ 𝑘 ) |
105 |
91
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑁 ∈ ℝ ) |
106 |
|
peano2rem |
⊢ ( 𝐽 ∈ ℝ → ( 𝐽 − 1 ) ∈ ℝ ) |
107 |
27 106
|
syl |
⊢ ( 𝜑 → ( 𝐽 − 1 ) ∈ ℝ ) |
108 |
107
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐽 − 1 ) ∈ ℝ ) |
109 |
|
elfzle2 |
⊢ ( 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) → 𝑘 ≤ ( 𝐽 − 1 ) ) |
110 |
109
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ≤ ( 𝐽 − 1 ) ) |
111 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝐽 ∈ ℝ ) |
112 |
111
|
ltm1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐽 − 1 ) < 𝐽 ) |
113 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝐽 ≤ 𝑁 ) |
114 |
108 111 105 112 113
|
ltletrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐽 − 1 ) < 𝑁 ) |
115 |
95 108 105 110 114
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 < 𝑁 ) |
116 |
95 105 115
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ≤ 𝑁 ) |
117 |
90 91 93 104 116
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
118 |
117 1
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
119 |
|
peano2zm |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 − 1 ) ∈ ℤ ) |
120 |
91 119
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ℤ ) |
121 |
120
|
zred |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ℝ ) |
122 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
123 |
27 28 122 30
|
lesub1dd |
⊢ ( 𝜑 → ( 𝐽 − 1 ) ≤ ( 𝑁 − 1 ) ) |
124 |
123
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐽 − 1 ) ≤ ( 𝑁 − 1 ) ) |
125 |
95 108 121 110 124
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ≤ ( 𝑁 − 1 ) ) |
126 |
90 120 93 104 125
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) |
127 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) |
128 |
|
fzoval |
⊢ ( 𝑁 ∈ ℤ → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) ) |
129 |
24 128
|
syl |
⊢ ( 𝜑 → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) ) |
130 |
129
|
eqcomd |
⊢ ( 𝜑 → ( 𝑀 ... ( 𝑁 − 1 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
131 |
130
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝑀 ... ( 𝑁 − 1 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
132 |
127 131
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) ) |
133 |
|
fzofzp1 |
⊢ ( 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
134 |
132 133
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
135 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → 𝜑 ) |
136 |
135 134
|
jca |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
137 |
|
eleq1 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝑗 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
138 |
137
|
anbi2d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
139 |
|
fveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
140 |
139
|
eleq1d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝑗 ) ∈ ℝ ↔ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) ) |
141 |
138 140
|
imbi12d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) ) ) |
142 |
|
eleq1 |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) ) |
143 |
142
|
anbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
144 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
145 |
144
|
eleq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) ) |
146 |
143 145
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) ) ) |
147 |
146 1
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) |
148 |
141 147
|
vtoclg |
⊢ ( ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) ) |
149 |
134 136 148
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
150 |
126 149
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
151 |
132 2
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) < ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
152 |
126 151
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) < ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
153 |
118 150 152
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐼 + 1 ) ... ( 𝐽 − 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
154 |
65 89 153
|
monoord |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝐹 ‘ 𝐽 ) ) |
155 |
13 43 51 59 154
|
ltletrd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐼 ) < ( 𝐹 ‘ 𝐽 ) ) |