Step |
Hyp |
Ref |
Expression |
1 |
|
monoordxrv.1 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
monoordxrv.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
3 |
|
monoordxrv.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
4 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
5 |
1 4
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
6 |
|
eleq1 |
⊢ ( 𝑥 = 𝑀 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑀 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑀 ) ) |
8 |
7
|
breq2d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) ) |
9 |
6 8
|
imbi12d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝜑 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝜑 → ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) ) ) ) |
11 |
|
eleq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 𝑛 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑛 ) ) |
13 |
12
|
breq2d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) ) |
14 |
11 13
|
imbi12d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝜑 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝜑 → ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
16 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
17 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
18 |
17
|
breq2d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
19 |
16 18
|
imbi12d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑥 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝜑 → ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
21 |
|
eleq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑁 ) ) |
23 |
22
|
breq2d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝜑 → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝜑 → ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) ) ) ) |
26 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
27 |
1 26
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
28 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
29 |
|
fveq2 |
⊢ ( 𝑘 = 𝑀 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑀 ) ) |
30 |
29
|
eleq1d |
⊢ ( 𝑘 = 𝑀 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ↔ ( 𝐹 ‘ 𝑀 ) ∈ ℝ* ) ) |
31 |
30
|
rspcv |
⊢ ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ* → ( 𝐹 ‘ 𝑀 ) ∈ ℝ* ) ) |
32 |
27 28 31
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ∈ ℝ* ) |
33 |
32
|
xrleidd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) |
34 |
33
|
a1d |
⊢ ( 𝜑 → ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) ) |
35 |
34
|
a1i |
⊢ ( 𝑀 ∈ ℤ → ( 𝜑 → ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑀 ) ) ) ) |
36 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
37 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
38 |
|
peano2fzr |
⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) |
39 |
36 37 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) |
40 |
39
|
expr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → 𝑛 ∈ ( 𝑀 ... 𝑁 ) ) ) |
41 |
40
|
imim1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) → ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) ) ) |
42 |
|
eluzelz |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑛 ∈ ℤ ) |
43 |
36 42
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → 𝑛 ∈ ℤ ) |
44 |
|
elfzuz3 |
⊢ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) |
45 |
37 44
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) |
46 |
|
eluzp1m1 |
⊢ ( ( 𝑛 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑛 ) ) |
47 |
43 45 46
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑛 ) ) |
48 |
|
elfzuzb |
⊢ ( 𝑛 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑛 ) ) ) |
49 |
36 47 48
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → 𝑛 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ) |
50 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
51 |
50
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ∀ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
52 |
|
fveq2 |
⊢ ( 𝑘 = 𝑛 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑛 ) ) |
53 |
|
fvoveq1 |
⊢ ( 𝑘 = 𝑛 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
54 |
52 53
|
breq12d |
⊢ ( 𝑘 = 𝑛 → ( ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐹 ‘ 𝑛 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
55 |
54
|
rspcv |
⊢ ( 𝑛 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) → ( ∀ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ( 𝐹 ‘ 𝑘 ) ≤ ( 𝐹 ‘ ( 𝑘 + 1 ) ) → ( 𝐹 ‘ 𝑛 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
56 |
49 51 55
|
sylc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝐹 ‘ 𝑛 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
57 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝐹 ‘ 𝑀 ) ∈ ℝ* ) |
58 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
59 |
52
|
eleq1d |
⊢ ( 𝑘 = 𝑛 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ↔ ( 𝐹 ‘ 𝑛 ) ∈ ℝ* ) ) |
60 |
59
|
rspcv |
⊢ ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ* → ( 𝐹 ‘ 𝑛 ) ∈ ℝ* ) ) |
61 |
39 58 60
|
sylc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ ℝ* ) |
62 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
63 |
62
|
eleq1d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ↔ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℝ* ) ) |
64 |
63
|
rspcv |
⊢ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ* → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℝ* ) ) |
65 |
37 58 64
|
sylc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℝ* ) |
66 |
|
xrletr |
⊢ ( ( ( 𝐹 ‘ 𝑀 ) ∈ ℝ* ∧ ( 𝐹 ‘ 𝑛 ) ∈ ℝ* ∧ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℝ* ) → ( ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ∧ ( 𝐹 ‘ 𝑛 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
67 |
57 61 65 66
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ∧ ( 𝐹 ‘ 𝑛 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
68 |
56 67
|
mpan2d |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
69 |
41 68
|
animpimp2impd |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝜑 → ( 𝑛 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑛 ) ) ) → ( 𝜑 → ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
70 |
10 15 20 25 35 69
|
uzind4 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝜑 → ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) ) ) |
71 |
1 70
|
mpcom |
⊢ ( 𝜑 → ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) ) |
72 |
5 71
|
mpd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ≤ ( 𝐹 ‘ 𝑁 ) ) |