Proof of Theorem xrralrecnnge
| Step | Hyp | Ref
| Expression |
| 1 | | xrralrecnnge.n |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
| 2 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑛 𝐴 ≤ 𝐵 |
| 3 | 1, 2 | nfan 1899 |
. . . 4
⊢
Ⅎ𝑛(𝜑 ∧ 𝐴 ≤ 𝐵) |
| 4 | | xrralrecnnge.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 5 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
| 6 | | nnrecre 12308 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
| 7 | 6 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ) |
| 8 | 5, 7 | resubcld 11691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈ ℝ) |
| 9 | 8 | rexrd 11311 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈
ℝ*) |
| 10 | 9 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ∈
ℝ*) |
| 11 | | xrralrecnnge.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 12 | 11 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈
ℝ*) |
| 13 | 4 | rexrd 11311 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 14 | 13 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈
ℝ*) |
| 15 | | nnrp 13046 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 16 | 15 | rpreccld 13087 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ+) |
| 17 | 16 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
| 18 | 5, 17 | ltsubrpd 13109 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴) |
| 19 | 18 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐴) |
| 20 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐵) |
| 21 | 10, 14, 12, 19, 20 | xrltletrd 13203 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) < 𝐵) |
| 22 | 10, 12, 21 | xrltled 13192 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐴 − (1 / 𝑛)) ≤ 𝐵) |
| 23 | 22 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑛 ∈ ℕ → (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
| 24 | 3, 23 | ralrimi 3257 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) |
| 25 | 24 | ex 412 |
. 2
⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
| 26 | | pnfxr 11315 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
| 27 | 26 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 28 | 4 | ltpnfd 13163 |
. . . . . . 7
⊢ (𝜑 → 𝐴 < +∞) |
| 29 | 13, 27, 28 | xrltled 13192 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ +∞) |
| 30 | 29 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ +∞) |
| 31 | | id 22 |
. . . . . . 7
⊢ (𝐵 = +∞ → 𝐵 = +∞) |
| 32 | 31 | eqcomd 2743 |
. . . . . 6
⊢ (𝐵 = +∞ → +∞ =
𝐵) |
| 33 | 32 | adantl 481 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → +∞ = 𝐵) |
| 34 | 30, 33 | breqtrd 5169 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵) |
| 35 | 11 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈
ℝ*) |
| 36 | | 1nn 12277 |
. . . . . . . . . . . . . 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 7439 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1)) |
| 40 | 39 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 1 → (𝐴 − (1 / 𝑛)) = (𝐴 − (1 / 1))) |
| 41 | 40 | breq1d 5153 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 1 → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ (𝐴 − (1 / 1)) ≤ 𝐵)) |
| 42 | 41 | rspcva 3620 |
. . . . . . . . . . . . 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 480 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤ 𝐵) |
| 45 | | simpr 484 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → 𝐵 = -∞) |
| 46 | 44, 45 | breqtrd 5169 |
. . . . . . . . . 10
⊢
((∀𝑛 ∈
ℕ (𝐴 − (1 /
𝑛)) ≤ 𝐵 ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤
-∞) |
| 47 | 46 | adantll 714 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → (𝐴 − (1 / 1)) ≤
-∞) |
| 48 | | 1red 11262 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ∈
ℝ) |
| 49 | | ax-1ne0 11224 |
. . . . . . . . . . . . . . 15
⊢ 1 ≠
0 |
| 50 | 49 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ≠ 0) |
| 51 | 48, 48, 50 | redivcld 12095 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 / 1) ∈
ℝ) |
| 52 | 4, 51 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − (1 / 1)) ∈
ℝ) |
| 53 | 52 | mnfltd 13166 |
. . . . . . . . . . 11
⊢ (𝜑 → -∞ < (𝐴 − (1 /
1))) |
| 54 | | mnfxr 11318 |
. . . . . . . . . . . . 13
⊢ -∞
∈ ℝ* |
| 55 | 54 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → -∞ ∈
ℝ*) |
| 56 | 52 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − (1 / 1)) ∈
ℝ*) |
| 57 | 55, 56 | xrltnled 45374 |
. . . . . . . . . . 11
⊢ (𝜑 → (-∞ < (𝐴 − (1 / 1)) ↔ ¬
(𝐴 − (1 / 1)) ≤
-∞)) |
| 58 | 53, 57 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → ¬ (𝐴 − (1 / 1)) ≤
-∞) |
| 59 | 58 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ 𝐵 = -∞) → ¬ (𝐴 − (1 / 1)) ≤
-∞) |
| 60 | 47, 59 | pm2.65da 817 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → ¬ 𝐵 = -∞) |
| 61 | 60 | neqned 2947 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐵 ≠ -∞) |
| 62 | 61 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ -∞) |
| 63 | | neqne 2948 |
. . . . . . 7
⊢ (¬
𝐵 = +∞ → 𝐵 ≠ +∞) |
| 64 | 63 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ +∞) |
| 65 | 35, 62, 64 | xrred 45376 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ) |
| 66 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛 𝐵 ∈ ℝ |
| 67 | 1, 66 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝜑 ∧ 𝐵 ∈ ℝ) |
| 68 | 13 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐴 ∈
ℝ*) |
| 69 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) |
| 70 | 67, 68, 69 | xrralrecnnle 45394 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 71 | 5 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
| 72 | 6 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ) |
| 73 | 69 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ) |
| 74 | 71, 72, 73 | lesubaddd 11860 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 75 | 74 | bicomd 223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
| 76 | 67, 75 | ralbida 3270 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |
| 77 | 70, 76 | bitr2d 280 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 ↔ 𝐴 ≤ 𝐵)) |
| 78 | 77 | biimpd 229 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵)) |
| 79 | 78 | imp 406 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐵 ∈ ℝ) ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
| 80 | 79 | an32s 652 |
. . . . 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 813 |
. . 3
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
| 83 | 82 | ex 412 |
. 2
⊢ (𝜑 → (∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵 → 𝐴 ≤ 𝐵)) |
| 84 | 25, 83 | impbid 212 |
1
⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) |