Proof of Theorem xrralrecnnge
Step | Hyp | Ref
| Expression |
1 | | xrralrecnnge.n |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
2 | | nfv 1917 |
. . . . 5
⊢
Ⅎ𝑛 𝐴 ≤ 𝐵 |
3 | 1, 2 | nfan 1902 |
. . . 4
⊢
Ⅎ𝑛(𝜑 ∧ 𝐴 ≤ 𝐵) |
4 | | xrralrecnnge.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
6 | | nnrecre 12015 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
7 | 6 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ) |
8 | 5, 7 | resubcld 11403 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈ ℝ) |
9 | 8 | rexrd 11025 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈
ℝ*) |
10 | 9 | adantlr 712 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈
ℝ*) |
11 | | xrralrecnnge.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
12 | 11 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈
ℝ*) |
13 | 4 | rexrd 11025 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
14 | 13 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈
ℝ*) |
15 | | nnrp 12741 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
16 | 15 | rpreccld 12782 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ+) |
17 | 16 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
18 | 5, 17 | ltsubrpd 12804 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴) |
19 | 18 | adantlr 712 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴) |
20 | | simplr 766 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐵) |
21 | 10, 14, 12, 19, 20 | xrltletrd 12895 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐵) |
22 | 10, 12, 21 | xrltled 12884 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ≤ 𝐵) |
23 | 22 | ex 413 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑛 ∈ ℕ → (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
24 | 3, 23 | ralrimi 3141 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) |
25 | 24 | ex 413 |
. 2
⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
26 | | pnfxr 11029 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
27 | 26 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → +∞ ∈
ℝ*) |
28 | 4 | ltpnfd 12857 |
. . . . . . 7
⊢ (𝜑 → 𝐴 < +∞) |
29 | 13, 27, 28 | xrltled 12884 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ +∞) |
30 | 29 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ +∞) |
31 | | id 22 |
. . . . . . 7
⊢ (𝐵 = +∞ → 𝐵 = +∞) |
32 | 31 | eqcomd 2744 |
. . . . . 6
⊢ (𝐵 = +∞ → +∞ =
𝐵) |
33 | 32 | adantl 482 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → +∞ = 𝐵) |
34 | 30, 33 | breqtrd 5100 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵) |
35 | 11 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈
ℝ*) |
36 | | 1nn 11984 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℕ |
37 | 36 | a1i 11 |
. . . . . . . . . . . . 13
⊢
(∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 → 1 ∈ ℕ) |
38 | | id 22 |
. . . . . . . . . . . . 13
⊢
(∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) |
39 | | oveq2 7283 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1)) |
40 | 39 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 1 → (𝐴 − (1 / 𝑛)) = (𝐴 − (1 / 1))) |
41 | 40 | breq1d 5084 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 1 → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ (𝐴 − (1 / 1)) ≤ 𝐵)) |
42 | 41 | rspcva 3559 |
. . . . . . . . . . . . 13
⊢ ((1
∈ ℕ ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → (𝐴 − (1 / 1)) ≤ 𝐵) |
43 | 37, 38, 42 | syl2anc 584 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 → (𝐴 − (1 / 1)) ≤ 𝐵) |
44 | 43 | adantr 481 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤ 𝐵) |
45 | | simpr 485 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → 𝐵 = -∞) |
46 | 44, 45 | breqtrd 5100 |
. . . . . . . . . 10
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤
-∞) |
47 | 46 | adantll 711 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤
-∞) |
48 | | 1red 10976 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ∈
ℝ) |
49 | | ax-1ne0 10940 |
. . . . . . . . . . . . . . 15
⊢ 1 ≠
0 |
50 | 49 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ≠ 0) |
51 | 48, 48, 50 | redivcld 11803 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 / 1) ∈
ℝ) |
52 | 4, 51 | resubcld 11403 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − (1 / 1)) ∈
ℝ) |
53 | 52 | mnfltd 12860 |
. . . . . . . . . . 11
⊢ (𝜑 → -∞ < (𝐴 − (1 /
1))) |
54 | | mnfxr 11032 |
. . . . . . . . . . . . 13
⊢ -∞
∈ ℝ* |
55 | 54 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → -∞ ∈
ℝ*) |
56 | 52 | rexrd 11025 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − (1 / 1)) ∈
ℝ*) |
57 | 55, 56 | xrltnled 42902 |
. . . . . . . . . . 11
⊢ (𝜑 → (-∞ < (𝐴 − (1 / 1)) ↔ ¬
(𝐴 − (1 / 1)) ≤
-∞)) |
58 | 53, 57 | mpbid 231 |
. . . . . . . . . 10
⊢ (𝜑 → ¬ (𝐴 − (1 / 1)) ≤
-∞) |
59 | 58 | ad2antrr 723 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → ¬ (𝐴 − (1 / 1)) ≤
-∞) |
60 | 47, 59 | pm2.65da 814 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → ¬ 𝐵 = -∞) |
61 | 60 | neqned 2950 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐵 ≠ -∞) |
62 | 61 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ -∞) |
63 | | neqne 2951 |
. . . . . . 7
⊢ (¬
𝐵 = +∞ → 𝐵 ≠ +∞) |
64 | 63 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞) |
65 | 35, 62, 64 | xrred 42904 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ) |
66 | | nfv 1917 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛 𝐵 ∈ ℝ |
67 | 1, 66 | nfan 1902 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝜑 ∧ 𝐵 ∈ ℝ) |
68 | 13 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈
ℝ*) |
69 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) |
70 | 67, 68, 69 | xrralrecnnle 42922 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
71 | 5 | adantlr 712 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
72 | 6 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ) |
73 | 69 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ) |
74 | 71, 72, 73 | lesubaddd 11572 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
75 | 74 | bicomd 222 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
76 | 67, 75 | ralbida 3159 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
77 | 70, 76 | bitr2d 279 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ 𝐵)) |
78 | 77 | biimpd 228 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵)) |
79 | 78 | imp 407 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
80 | 79 | an32s 649 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ 𝐵) |
81 | 65, 80 | syldan 591 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐴 ≤ 𝐵) |
82 | 34, 81 | pm2.61dan 810 |
. . 3
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
83 | 82 | ex 413 |
. 2
⊢ (𝜑 → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵)) |
84 | 25, 83 | impbid 211 |
1
⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |