Step |
Hyp |
Ref |
Expression |
1 |
|
ssinc.1 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
ssinc.2 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) |
3 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
7 |
4 6
|
jca |
⊢ ( 𝜑 → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
8 |
|
eluzle |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ 𝑁 ) |
9 |
1 8
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ 𝑁 ) |
10 |
6
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
11 |
10
|
leidd |
⊢ ( 𝜑 → 𝑁 ≤ 𝑁 ) |
12 |
6 9 11
|
3jca |
⊢ ( 𝜑 → ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) ) |
13 |
7 12
|
jca |
⊢ ( 𝜑 → ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) ) ) |
14 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
15 |
|
fveq2 |
⊢ ( 𝑛 = 𝑀 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑀 ) ) |
16 |
15
|
sseq2d |
⊢ ( 𝑛 = 𝑀 → ( ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) |
17 |
16
|
imbi2d |
⊢ ( 𝑛 = 𝑀 → ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑚 ) ) |
19 |
18
|
sseq2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) |
22 |
21
|
sseq2d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) ) ) |
24 |
|
fveq2 |
⊢ ( 𝑛 = 𝑁 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑁 ) ) |
25 |
24
|
sseq2d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑁 ) ) ) |
26 |
25
|
imbi2d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑁 ) ) ) ) |
27 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) |
28 |
27
|
a1i |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) |
29 |
|
simpr |
⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → 𝜑 ) |
30 |
|
simpl |
⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ) |
31 |
|
pm3.35 |
⊢ ( ( 𝜑 ∧ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ) → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) |
32 |
29 30 31
|
syl2anc |
⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) |
33 |
32
|
3adant1 |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) |
34 |
|
simpr |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝜑 ) |
35 |
|
simplll |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑀 ∈ ℤ ) |
36 |
|
simplr1 |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ℤ ) |
37 |
|
simplr2 |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑀 ≤ 𝑚 ) |
38 |
35 36 37
|
3jca |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) ) |
39 |
|
eluz2 |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) ) |
40 |
38 39
|
sylibr |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
41 |
|
simpllr |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑁 ∈ ℤ ) |
42 |
|
simplr3 |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 < 𝑁 ) |
43 |
40 41 42
|
3jca |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ∧ 𝑚 < 𝑁 ) ) |
44 |
|
elfzo2 |
⊢ ( 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ∧ 𝑚 < 𝑁 ) ) |
45 |
43 44
|
sylibr |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) |
46 |
34 45 2
|
syl2anc |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) |
47 |
46
|
3adant2 |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) |
48 |
33 47
|
sstrd |
⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) |
49 |
48
|
3exp |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑚 ) ) → ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ) ) ) |
50 |
17 20 23 26 28 49
|
fzind |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) ) → ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑁 ) ) ) |
51 |
13 14 50
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑁 ) ) |