Proof of Theorem pntibndlem1
| Step | Hyp | Ref
| Expression |
| 1 | | pntibndlem1.l |
. . . 4
⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) |
| 2 | | 4nn 12349 |
. . . . . 6
⊢ 4 ∈
ℕ |
| 3 | | nnrp 13046 |
. . . . . 6
⊢ (4 ∈
ℕ → 4 ∈ ℝ+) |
| 4 | | rpreccl 13061 |
. . . . . 6
⊢ (4 ∈
ℝ+ → (1 / 4) ∈ ℝ+) |
| 5 | 2, 3, 4 | mp2b 10 |
. . . . 5
⊢ (1 / 4)
∈ ℝ+ |
| 6 | | pntibndlem1.1 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
| 7 | | 3rp 13040 |
. . . . . 6
⊢ 3 ∈
ℝ+ |
| 8 | | rpaddcl 13057 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ+
∧ 3 ∈ ℝ+) → (𝐴 + 3) ∈
ℝ+) |
| 9 | 6, 7, 8 | sylancl 586 |
. . . . 5
⊢ (𝜑 → (𝐴 + 3) ∈
ℝ+) |
| 10 | | rpdivcl 13060 |
. . . . 5
⊢ (((1 / 4)
∈ ℝ+ ∧ (𝐴 + 3) ∈ ℝ+) → ((1
/ 4) / (𝐴 + 3)) ∈
ℝ+) |
| 11 | 5, 9, 10 | sylancr 587 |
. . . 4
⊢ (𝜑 → ((1 / 4) / (𝐴 + 3)) ∈
ℝ+) |
| 12 | 1, 11 | eqeltrid 2845 |
. . 3
⊢ (𝜑 → 𝐿 ∈
ℝ+) |
| 13 | 12 | rpred 13077 |
. 2
⊢ (𝜑 → 𝐿 ∈ ℝ) |
| 14 | 12 | rpgt0d 13080 |
. 2
⊢ (𝜑 → 0 < 𝐿) |
| 15 | | rpcn 13045 |
. . . . . . 7
⊢ ((1 / 4)
∈ ℝ+ → (1 / 4) ∈ ℂ) |
| 16 | 5, 15 | ax-mp 5 |
. . . . . 6
⊢ (1 / 4)
∈ ℂ |
| 17 | 16 | div1i 11995 |
. . . . 5
⊢ ((1 / 4)
/ 1) = (1 / 4) |
| 18 | | rpre 13043 |
. . . . . . 7
⊢ ((1 / 4)
∈ ℝ+ → (1 / 4) ∈ ℝ) |
| 19 | 5, 18 | mp1i 13 |
. . . . . 6
⊢ (𝜑 → (1 / 4) ∈
ℝ) |
| 20 | | 3re 12346 |
. . . . . . 7
⊢ 3 ∈
ℝ |
| 21 | 20 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 3 ∈
ℝ) |
| 22 | 9 | rpred 13077 |
. . . . . 6
⊢ (𝜑 → (𝐴 + 3) ∈ ℝ) |
| 23 | | 1lt4 12442 |
. . . . . . . . 9
⊢ 1 <
4 |
| 24 | | 4re 12350 |
. . . . . . . . . 10
⊢ 4 ∈
ℝ |
| 25 | | 4pos 12373 |
. . . . . . . . . 10
⊢ 0 <
4 |
| 26 | | recgt1 12164 |
. . . . . . . . . 10
⊢ ((4
∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) <
1)) |
| 27 | 24, 25, 26 | mp2an 692 |
. . . . . . . . 9
⊢ (1 < 4
↔ (1 / 4) < 1) |
| 28 | 23, 27 | mpbi 230 |
. . . . . . . 8
⊢ (1 / 4)
< 1 |
| 29 | | 1lt3 12439 |
. . . . . . . 8
⊢ 1 <
3 |
| 30 | 5, 18 | ax-mp 5 |
. . . . . . . . 9
⊢ (1 / 4)
∈ ℝ |
| 31 | | 1re 11261 |
. . . . . . . . 9
⊢ 1 ∈
ℝ |
| 32 | 30, 31, 20 | lttri 11387 |
. . . . . . . 8
⊢ (((1 / 4)
< 1 ∧ 1 < 3) → (1 / 4) < 3) |
| 33 | 28, 29, 32 | mp2an 692 |
. . . . . . 7
⊢ (1 / 4)
< 3 |
| 34 | 33 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (1 / 4) <
3) |
| 35 | | ltaddrp 13072 |
. . . . . . . 8
⊢ ((3
∈ ℝ ∧ 𝐴
∈ ℝ+) → 3 < (3 + 𝐴)) |
| 36 | 20, 6, 35 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → 3 < (3 + 𝐴)) |
| 37 | | 3cn 12347 |
. . . . . . . 8
⊢ 3 ∈
ℂ |
| 38 | 6 | rpcnd 13079 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 39 | | addcom 11447 |
. . . . . . . 8
⊢ ((3
∈ ℂ ∧ 𝐴
∈ ℂ) → (3 + 𝐴) = (𝐴 + 3)) |
| 40 | 37, 38, 39 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (3 + 𝐴) = (𝐴 + 3)) |
| 41 | 36, 40 | breqtrd 5169 |
. . . . . 6
⊢ (𝜑 → 3 < (𝐴 + 3)) |
| 42 | 19, 21, 22, 34, 41 | lttrd 11422 |
. . . . 5
⊢ (𝜑 → (1 / 4) < (𝐴 + 3)) |
| 43 | 17, 42 | eqbrtrid 5178 |
. . . 4
⊢ (𝜑 → ((1 / 4) / 1) < (𝐴 + 3)) |
| 44 | 31 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ) |
| 45 | | 0lt1 11785 |
. . . . . 6
⊢ 0 <
1 |
| 46 | 45 | a1i 11 |
. . . . 5
⊢ (𝜑 → 0 < 1) |
| 47 | 9 | rpregt0d 13083 |
. . . . 5
⊢ (𝜑 → ((𝐴 + 3) ∈ ℝ ∧ 0 < (𝐴 + 3))) |
| 48 | | ltdiv23 12159 |
. . . . 5
⊢ (((1 / 4)
∈ ℝ ∧ (1 ∈ ℝ ∧ 0 < 1) ∧ ((𝐴 + 3) ∈ ℝ ∧ 0
< (𝐴 + 3))) → (((1
/ 4) / 1) < (𝐴 + 3)
↔ ((1 / 4) / (𝐴 + 3))
< 1)) |
| 49 | 19, 44, 46, 47, 48 | syl121anc 1377 |
. . . 4
⊢ (𝜑 → (((1 / 4) / 1) < (𝐴 + 3) ↔ ((1 / 4) / (𝐴 + 3)) < 1)) |
| 50 | 43, 49 | mpbid 232 |
. . 3
⊢ (𝜑 → ((1 / 4) / (𝐴 + 3)) < 1) |
| 51 | 1, 50 | eqbrtrid 5178 |
. 2
⊢ (𝜑 → 𝐿 < 1) |
| 52 | | 0xr 11308 |
. . 3
⊢ 0 ∈
ℝ* |
| 53 | | 1xr 11320 |
. . 3
⊢ 1 ∈
ℝ* |
| 54 | | elioo2 13428 |
. . 3
⊢ ((0
∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐿 ∈ (0(,)1) ↔ (𝐿 ∈ ℝ ∧ 0 <
𝐿 ∧ 𝐿 < 1))) |
| 55 | 52, 53, 54 | mp2an 692 |
. 2
⊢ (𝐿 ∈ (0(,)1) ↔ (𝐿 ∈ ℝ ∧ 0 <
𝐿 ∧ 𝐿 < 1)) |
| 56 | 13, 14, 51, 55 | syl3anbrc 1344 |
1
⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |