Step |
Hyp |
Ref |
Expression |
1 |
|
iooiinicc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
iooiinicc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝐴 ∈ ℝ ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝐵 ∈ ℝ ) |
5 |
|
1nn |
⊢ 1 ∈ ℕ |
6 |
|
ioossre |
⊢ ( ( 𝐴 − ( 1 / 1 ) ) (,) ( 𝐵 + ( 1 / 1 ) ) ) ⊆ ℝ |
7 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 1 / 𝑛 ) = ( 1 / 1 ) ) |
8 |
7
|
oveq2d |
⊢ ( 𝑛 = 1 → ( 𝐴 − ( 1 / 𝑛 ) ) = ( 𝐴 − ( 1 / 1 ) ) ) |
9 |
7
|
oveq2d |
⊢ ( 𝑛 = 1 → ( 𝐵 + ( 1 / 𝑛 ) ) = ( 𝐵 + ( 1 / 1 ) ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑛 = 1 → ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) = ( ( 𝐴 − ( 1 / 1 ) ) (,) ( 𝐵 + ( 1 / 1 ) ) ) ) |
11 |
10
|
sseq1d |
⊢ ( 𝑛 = 1 → ( ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ ↔ ( ( 𝐴 − ( 1 / 1 ) ) (,) ( 𝐵 + ( 1 / 1 ) ) ) ⊆ ℝ ) ) |
12 |
11
|
rspcev |
⊢ ( ( 1 ∈ ℕ ∧ ( ( 𝐴 − ( 1 / 1 ) ) (,) ( 𝐵 + ( 1 / 1 ) ) ) ⊆ ℝ ) → ∃ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ ) |
13 |
5 6 12
|
mp2an |
⊢ ∃ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ |
14 |
|
iinss |
⊢ ( ∃ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ ) |
15 |
13 14
|
ax-mp |
⊢ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ |
16 |
15
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ℝ ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
18 |
16 17
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 ∈ ℝ ) |
19 |
|
nfv |
⊢ Ⅎ 𝑛 𝜑 |
20 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑥 |
21 |
|
nfii1 |
⊢ Ⅎ 𝑛 ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) |
22 |
20 21
|
nfel |
⊢ Ⅎ 𝑛 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) |
23 |
19 22
|
nfan |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
24 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝜑 ) |
25 |
|
iinss2 |
⊢ ( 𝑛 ∈ ℕ → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
26 |
25
|
adantl |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
27 |
|
simpl |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
28 |
26 27
|
sseldd |
⊢ ( ( 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
29 |
28
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
30 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
31 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 ∈ ℝ ) |
32 |
31
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 ∈ ℝ ) |
33 |
|
elioore |
⊢ ( 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) → 𝑥 ∈ ℝ ) |
34 |
33
|
adantr |
⊢ ( ( 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ℝ ) |
35 |
|
nnrecre |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℝ ) |
36 |
35
|
adantl |
⊢ ( ( 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ ) |
37 |
34 36
|
readdcld |
⊢ ( ( 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑥 + ( 1 / 𝑛 ) ) ∈ ℝ ) |
38 |
37
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑥 + ( 1 / 𝑛 ) ) ∈ ℝ ) |
39 |
35
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ ) |
40 |
31 39
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ ) |
41 |
40
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ* ) |
42 |
41
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ* ) |
43 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐵 ∈ ℝ ) |
44 |
43 39
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ ) |
45 |
44
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ* ) |
46 |
45
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ* ) |
47 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
48 |
|
ioogtlb |
⊢ ( ( ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ* ∧ ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝐴 − ( 1 / 𝑛 ) ) < 𝑥 ) |
49 |
42 46 47 48
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 − ( 1 / 𝑛 ) ) < 𝑥 ) |
50 |
35
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ ) |
51 |
34
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ ℝ ) |
52 |
32 50 51
|
ltsubaddd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 − ( 1 / 𝑛 ) ) < 𝑥 ↔ 𝐴 < ( 𝑥 + ( 1 / 𝑛 ) ) ) ) |
53 |
49 52
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 < ( 𝑥 + ( 1 / 𝑛 ) ) ) |
54 |
32 38 53
|
ltled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 ≤ ( 𝑥 + ( 1 / 𝑛 ) ) ) |
55 |
24 29 30 54
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 ≤ ( 𝑥 + ( 1 / 𝑛 ) ) ) |
56 |
55
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝑛 ∈ ℕ → 𝐴 ≤ ( 𝑥 + ( 1 / 𝑛 ) ) ) ) |
57 |
23 56
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ∀ 𝑛 ∈ ℕ 𝐴 ≤ ( 𝑥 + ( 1 / 𝑛 ) ) ) |
58 |
3
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝐴 ∈ ℝ* ) |
59 |
23 58 18
|
xrralrecnnle |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝐴 ≤ 𝑥 ↔ ∀ 𝑛 ∈ ℕ 𝐴 ≤ ( 𝑥 + ( 1 / 𝑛 ) ) ) ) |
60 |
57 59
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝐴 ≤ 𝑥 ) |
61 |
44
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ ) |
62 |
|
iooltub |
⊢ ( ( ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ* ∧ ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 < ( 𝐵 + ( 1 / 𝑛 ) ) ) |
63 |
42 46 47 62
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 < ( 𝐵 + ( 1 / 𝑛 ) ) ) |
64 |
51 61 63
|
ltled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ≤ ( 𝐵 + ( 1 / 𝑛 ) ) ) |
65 |
24 29 30 64
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ≤ ( 𝐵 + ( 1 / 𝑛 ) ) ) |
66 |
65
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝑛 ∈ ℕ → 𝑥 ≤ ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
67 |
23 66
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ∀ 𝑛 ∈ ℕ 𝑥 ≤ ( 𝐵 + ( 1 / 𝑛 ) ) ) |
68 |
18
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 ∈ ℝ* ) |
69 |
23 68 4
|
xrralrecnnle |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝑥 ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ 𝑥 ≤ ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
70 |
67 69
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 ≤ 𝐵 ) |
71 |
3 4 18 60 70
|
eliccd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
72 |
71
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
73 |
|
dfss3 |
⊢ ( ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ↔ ∀ 𝑥 ∈ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
74 |
72 73
|
sylibr |
⊢ ( 𝜑 → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
75 |
|
1rp |
⊢ 1 ∈ ℝ+ |
76 |
75
|
a1i |
⊢ ( 𝑛 ∈ ℕ → 1 ∈ ℝ+ ) |
77 |
|
nnrp |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ+ ) |
78 |
76 77
|
rpdivcld |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℝ+ ) |
79 |
78
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ+ ) |
80 |
31 79
|
ltsubrpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 − ( 1 / 𝑛 ) ) < 𝐴 ) |
81 |
43 79
|
ltaddrpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐵 < ( 𝐵 + ( 1 / 𝑛 ) ) ) |
82 |
|
iccssioo |
⊢ ( ( ( ( 𝐴 − ( 1 / 𝑛 ) ) ∈ ℝ* ∧ ( 𝐵 + ( 1 / 𝑛 ) ) ∈ ℝ* ) ∧ ( ( 𝐴 − ( 1 / 𝑛 ) ) < 𝐴 ∧ 𝐵 < ( 𝐵 + ( 1 / 𝑛 ) ) ) ) → ( 𝐴 [,] 𝐵 ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
83 |
41 45 80 81 82
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 [,] 𝐵 ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
84 |
83
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 𝐴 [,] 𝐵 ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
85 |
|
ssiin |
⊢ ( ( 𝐴 [,] 𝐵 ) ⊆ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ↔ ∀ 𝑛 ∈ ℕ ( 𝐴 [,] 𝐵 ) ⊆ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
86 |
84 85
|
sylibr |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) ) |
87 |
74 86
|
eqssd |
⊢ ( 𝜑 → ∩ 𝑛 ∈ ℕ ( ( 𝐴 − ( 1 / 𝑛 ) ) (,) ( 𝐵 + ( 1 / 𝑛 ) ) ) = ( 𝐴 [,] 𝐵 ) ) |