Step |
Hyp |
Ref |
Expression |
1 |
|
isercoll.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
isercoll.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
isercoll.g |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ 𝑍 ) |
4 |
|
isercoll.i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) |
5 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
6 |
1 5
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
7 |
|
zssre |
⊢ ℤ ⊆ ℝ |
8 |
6 7
|
sstri |
⊢ 𝑍 ⊆ ℝ |
9 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝐺 : ℕ ⟶ 𝑍 ) |
10 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℕ ) |
11 |
9 10
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝑍 ) |
12 |
8 11
|
sselid |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
13 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℕ ) |
14 |
13
|
nnred |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℝ ) |
15 |
12 14
|
resubcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) − 𝑦 ) ∈ ℝ ) |
16 |
10
|
nnred |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℝ ) |
17 |
12 16
|
resubcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ∈ ℝ ) |
18 |
9 13
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑍 ) |
19 |
8 18
|
sselid |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ ℝ ) |
20 |
19 14
|
resubcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ∈ ℝ ) |
21 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 < 𝑦 ) |
22 |
16 14 12 21
|
ltsub2dd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) − 𝑦 ) < ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ) |
23 |
10
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℤ ) |
24 |
13
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℤ ) |
25 |
16 14 21
|
ltled |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ≤ 𝑦 ) |
26 |
|
eluz2 |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑥 ) ↔ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑥 ≤ 𝑦 ) ) |
27 |
23 24 25 26
|
syl3anbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ( ℤ≥ ‘ 𝑥 ) ) |
28 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑥 ... 𝑦 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) |
29 |
|
eluznn |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) → 𝑘 ∈ ℕ ) |
30 |
10 29
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) → 𝑘 ∈ ℕ ) |
31 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑘 ) ) |
32 |
|
id |
⊢ ( 𝑛 = 𝑘 → 𝑛 = 𝑘 ) |
33 |
31 32
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) = ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ) |
34 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) |
35 |
|
ovex |
⊢ ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ∈ V |
36 |
33 34 35
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ) |
37 |
36
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ) |
38 |
9
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ 𝑍 ) |
39 |
8 38
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
40 |
|
nnre |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ ) |
41 |
40
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℝ ) |
42 |
39 41
|
resubcld |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ∈ ℝ ) |
43 |
37 42
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ ) |
44 |
30 43
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ ) |
45 |
28 44
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( 𝑥 ... 𝑦 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ ) |
46 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑥 ... ( 𝑦 − 1 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) |
47 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
48 |
|
ffvelrn |
⊢ ( ( 𝐺 : ℕ ⟶ 𝑍 ∧ ( 𝑘 + 1 ) ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ 𝑍 ) |
49 |
9 47 48
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ 𝑍 ) |
50 |
8 49
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℝ ) |
51 |
|
peano2rem |
⊢ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℝ → ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ∈ ℝ ) |
52 |
50 51
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ∈ ℝ ) |
53 |
4
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) |
54 |
6 38
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ ℤ ) |
55 |
6 49
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℤ ) |
56 |
|
zltlem1 |
⊢ ( ( ( 𝐺 ‘ 𝑘 ) ∈ ℤ ∧ ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℤ ) → ( ( 𝐺 ‘ 𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐺 ‘ 𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ) ) |
57 |
54 55 56
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐺 ‘ 𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ) ) |
58 |
53 57
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ) |
59 |
39 52 41 58
|
lesub1dd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ≤ ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) ) |
60 |
50
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
61 |
|
1cnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℂ ) |
62 |
41
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
63 |
60 61 62
|
sub32d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) = ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 𝑘 ) − 1 ) ) |
64 |
60 62 61
|
subsub4d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 𝑘 ) − 1 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
65 |
63 64
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
66 |
59 65
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑘 ) − 𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
67 |
47
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℕ ) |
68 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ ( 𝑘 + 1 ) ) ) |
69 |
|
id |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → 𝑛 = ( 𝑘 + 1 ) ) |
70 |
68 69
|
oveq12d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
71 |
|
ovex |
⊢ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ∈ V |
72 |
70 34 71
|
fvmpt |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
73 |
67 72
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ) |
74 |
66 37 73
|
3brtr4d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) ) |
75 |
30 74
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑥 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) ) |
76 |
46 75
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( 𝑥 ... ( 𝑦 − 1 ) ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) ) |
77 |
27 45 76
|
monoord |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) ) |
78 |
|
fveq2 |
⊢ ( 𝑛 = 𝑥 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑥 ) ) |
79 |
|
id |
⊢ ( 𝑛 = 𝑥 → 𝑛 = 𝑥 ) |
80 |
78 79
|
oveq12d |
⊢ ( 𝑛 = 𝑥 → ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) = ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ) |
81 |
|
ovex |
⊢ ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ∈ V |
82 |
80 34 81
|
fvmpt |
⊢ ( 𝑥 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) = ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ) |
83 |
10 82
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) = ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ) |
84 |
|
fveq2 |
⊢ ( 𝑛 = 𝑦 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑦 ) ) |
85 |
|
id |
⊢ ( 𝑛 = 𝑦 → 𝑛 = 𝑦 ) |
86 |
84 85
|
oveq12d |
⊢ ( 𝑛 = 𝑦 → ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) = ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) |
87 |
|
ovex |
⊢ ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ∈ V |
88 |
86 34 87
|
fvmpt |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) = ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) |
89 |
13 88
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺 ‘ 𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) = ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) |
90 |
77 83 89
|
3brtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) − 𝑥 ) ≤ ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) |
91 |
15 17 20 22 90
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) − 𝑦 ) < ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) |
92 |
12 19 14
|
ltsub1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ↔ ( ( 𝐺 ‘ 𝑥 ) − 𝑦 ) < ( ( 𝐺 ‘ 𝑦 ) − 𝑦 ) ) ) |
93 |
91 92
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) |
94 |
93
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) → ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) |
95 |
94
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) |
96 |
|
ss2ralv |
⊢ ( 𝑆 ⊆ ℕ → ( ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) ) |
97 |
95 96
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) |
98 |
|
nnssre |
⊢ ℕ ⊆ ℝ |
99 |
|
ltso |
⊢ < Or ℝ |
100 |
|
soss |
⊢ ( ℕ ⊆ ℝ → ( < Or ℝ → < Or ℕ ) ) |
101 |
98 99 100
|
mp2 |
⊢ < Or ℕ |
102 |
101
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → < Or ℕ ) |
103 |
|
soss |
⊢ ( 𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍 ) ) |
104 |
8 99 103
|
mp2 |
⊢ < Or 𝑍 |
105 |
104
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → < Or 𝑍 ) |
106 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → 𝐺 : ℕ ⟶ 𝑍 ) |
107 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → 𝑆 ⊆ ℕ ) |
108 |
|
soisores |
⊢ ( ( ( < Or ℕ ∧ < Or 𝑍 ) ∧ ( 𝐺 : ℕ ⟶ 𝑍 ∧ 𝑆 ⊆ ℕ ) ) → ( ( 𝐺 ↾ 𝑆 ) Isom < , < ( 𝑆 , ( 𝐺 “ 𝑆 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) ) |
109 |
102 105 106 107 108
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → ( ( 𝐺 ↾ 𝑆 ) Isom < , < ( 𝑆 , ( 𝐺 “ 𝑆 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → ( 𝐺 ‘ 𝑥 ) < ( 𝐺 ‘ 𝑦 ) ) ) ) |
110 |
97 109
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑆 ⊆ ℕ ) → ( 𝐺 ↾ 𝑆 ) Isom < , < ( 𝑆 , ( 𝐺 “ 𝑆 ) ) ) |