| Step | Hyp | Ref
| Expression |
| 1 | | xrralrecnnle.n |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
| 2 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑛 𝐴 ≤ 𝐵 |
| 3 | 1, 2 | nfan 1899 |
. . . 4
⊢
Ⅎ𝑛(𝜑 ∧ 𝐴 ≤ 𝐵) |
| 4 | | xrralrecnnle.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 5 | 4 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈
ℝ*) |
| 6 | | xrralrecnnle.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 7 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ) |
| 8 | | nnrecre 12308 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
| 9 | 8 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ) |
| 10 | 7, 9 | readdcld 11290 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ) |
| 11 | 10 | rexrd 11311 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈
ℝ*) |
| 12 | 11 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈
ℝ*) |
| 13 | | rexr 11307 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℝ → 𝐵 ∈
ℝ*) |
| 14 | 6, 13 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 15 | 14 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈
ℝ*) |
| 16 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐵) |
| 17 | | nnrp 13046 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 18 | | rpreccl 13061 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℝ+
→ (1 / 𝑛) ∈
ℝ+) |
| 19 | 17, 18 | syl 17 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ+) |
| 20 | 19 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
| 21 | 7, 20 | ltaddrpd 13110 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛))) |
| 22 | 21 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛))) |
| 23 | 5, 15, 12, 16, 22 | xrlelttrd 13202 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 < (𝐵 + (1 / 𝑛))) |
| 24 | 5, 12, 23 | xrltled 13192 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 25 | 24 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑛 ∈ ℕ → 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 26 | 3, 25 | ralrimi 3257 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 27 | 26 | ex 412 |
. 2
⊢ (𝜑 → (𝐴 ≤ 𝐵 → ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 28 | | rpgtrecnn 45391 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ ∃𝑛 ∈
ℕ (1 / 𝑛) < 𝑥) |
| 29 | 28 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) →
∃𝑛 ∈ ℕ (1
/ 𝑛) < 𝑥) |
| 30 | | nfra1 3284 |
. . . . . . . . 9
⊢
Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) |
| 31 | 1, 30 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 32 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑛 𝑥 ∈
ℝ+ |
| 33 | 31, 32 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) |
| 34 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑛 𝐴 ≤ (𝐵 + 𝑥) |
| 35 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝜑) |
| 36 | | rspa 3248 |
. . . . . . . . . . . 12
⊢
((∀𝑛 ∈
ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 37 | 36 | adantll 714 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 38 | 35, 37 | jca 511 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → (𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 39 | 38 | adantlr 715 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → (𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |
| 40 | | simplr 769 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈
ℝ+) |
| 41 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℕ) |
| 42 | 4 | ad4antr 732 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝐴 ∈
ℝ*) |
| 43 | 6 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈
ℝ) |
| 44 | | rpre 13043 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
| 45 | 44 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ) |
| 46 | 43, 45 | readdcld 11290 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐵 + 𝑥) ∈ ℝ) |
| 47 | 46 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐵 + 𝑥) ∈
ℝ*) |
| 48 | 47 | ad5ant13 757 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (𝐵 + 𝑥) ∈
ℝ*) |
| 49 | 11 | ad5ant14 758 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (𝐵 + (1 / 𝑛)) ∈
ℝ*) |
| 50 | | simp-4r 784 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝐴 ≤ (𝐵 + (1 / 𝑛))) |
| 51 | 8 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (1 / 𝑛) ∈ ℝ) |
| 52 | 45 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝑥 ∈ ℝ) |
| 53 | 43 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝐵 ∈ ℝ) |
| 54 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (1 / 𝑛) < 𝑥) |
| 55 | 51, 52, 53, 54 | ltadd2dd 11420 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (𝐵 + (1 / 𝑛)) < (𝐵 + 𝑥)) |
| 56 | 55 | adantl3r 750 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → (𝐵 + (1 / 𝑛)) < (𝐵 + 𝑥)) |
| 57 | 42, 49, 48, 50, 56 | xrlelttrd 13202 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝐴 < (𝐵 + 𝑥)) |
| 58 | 42, 48, 57 | xrltled 13192 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) ∧ (1 /
𝑛) < 𝑥) → 𝐴 ≤ (𝐵 + 𝑥)) |
| 59 | 58 | ex 412 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((1 /
𝑛) < 𝑥 → 𝐴 ≤ (𝐵 + 𝑥))) |
| 60 | 39, 40, 41, 59 | syl21anc 838 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((1 /
𝑛) < 𝑥 → 𝐴 ≤ (𝐵 + 𝑥))) |
| 61 | 60 | ex 412 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) → (𝑛 ∈ ℕ → ((1 /
𝑛) < 𝑥 → 𝐴 ≤ (𝐵 + 𝑥)))) |
| 62 | 33, 34, 61 | rexlimd 3266 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) →
(∃𝑛 ∈ ℕ (1
/ 𝑛) < 𝑥 → 𝐴 ≤ (𝐵 + 𝑥))) |
| 63 | 29, 62 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) ∧ 𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 + 𝑥)) |
| 64 | 63 | ralrimiva 3146 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) → ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥)) |
| 65 | | xralrple 13247 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+
𝐴 ≤ (𝐵 + 𝑥))) |
| 66 | 4, 6, 65 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) |
| 67 | 66 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) |
| 68 | 64, 67 | mpbird 257 |
. . 3
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))) → 𝐴 ≤ 𝐵) |
| 69 | 68 | ex 412 |
. 2
⊢ (𝜑 → (∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)) → 𝐴 ≤ 𝐵)) |
| 70 | 27, 69 | impbid 212 |
1
⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛)))) |