| Step | Hyp | Ref
| Expression |
| 1 | | xrnegiso.1 |
. . . . . 6
⊢ 𝐹 = (𝑥 ∈ ℝ* ↦
-𝑒𝑥) |
| 2 | | simpr 110 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ ℝ*) → 𝑥 ∈ ℝ*) |
| 3 | 2 | xnegcld 9930 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ ℝ*) → -𝑒𝑥 ∈ ℝ*) |
| 4 | | simpr 110 |
. . . . . . 7
⊢
((⊤ ∧ 𝑦
∈ ℝ*) → 𝑦 ∈ ℝ*) |
| 5 | 4 | xnegcld 9930 |
. . . . . 6
⊢
((⊤ ∧ 𝑦
∈ ℝ*) → -𝑒𝑦 ∈ ℝ*) |
| 6 | | xnegneg 9908 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ*
→ -𝑒-𝑒𝑥 = 𝑥) |
| 7 | 6 | eqeq2d 2208 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ*
→ (-𝑒𝑦 =
-𝑒-𝑒𝑥 ↔ -𝑒𝑦 = 𝑥)) |
| 8 | 7 | adantr 276 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (-𝑒𝑦 =
-𝑒-𝑒𝑥 ↔ -𝑒𝑦 = 𝑥)) |
| 9 | | eqcom 2198 |
. . . . . . . . 9
⊢
(-𝑒𝑦 = 𝑥 ↔ 𝑥 = -𝑒𝑦) |
| 10 | 8, 9 | bitrdi 196 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (-𝑒𝑦 =
-𝑒-𝑒𝑥 ↔ 𝑥 = -𝑒𝑦)) |
| 11 | | simpr 110 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → 𝑦 ∈ ℝ*) |
| 12 | | xnegcl 9907 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ*
→ -𝑒𝑥 ∈ ℝ*) |
| 13 | 12 | adantr 276 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → -𝑒𝑥 ∈ ℝ*) |
| 14 | | xneg11 9909 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ*
∧ -𝑒𝑥 ∈ ℝ*) →
(-𝑒𝑦 =
-𝑒-𝑒𝑥 ↔ 𝑦 = -𝑒𝑥)) |
| 15 | 11, 13, 14 | syl2anc 411 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (-𝑒𝑦 =
-𝑒-𝑒𝑥 ↔ 𝑦 = -𝑒𝑥)) |
| 16 | 10, 15 | bitr3d 190 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑥 = -𝑒𝑦 ↔ 𝑦 = -𝑒𝑥)) |
| 17 | 16 | adantl 277 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ* ∧ 𝑦 ∈ ℝ*)) → (𝑥 = -𝑒𝑦 ↔ 𝑦 = -𝑒𝑥)) |
| 18 | 1, 3, 5, 17 | f1ocnv2d 6127 |
. . . . 5
⊢ (⊤
→ (𝐹:ℝ*–1-1-onto→ℝ* ∧ ◡𝐹 = (𝑦 ∈ ℝ* ↦
-𝑒𝑦))) |
| 19 | 18 | mptru 1373 |
. . . 4
⊢ (𝐹:ℝ*–1-1-onto→ℝ* ∧ ◡𝐹 = (𝑦 ∈ ℝ* ↦
-𝑒𝑦)) |
| 20 | 19 | simpli 111 |
. . 3
⊢ 𝐹:ℝ*–1-1-onto→ℝ* |
| 21 | | simpl 109 |
. . . . . . 7
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → 𝑧 ∈ ℝ*) |
| 22 | 21 | xnegcld 9930 |
. . . . . 6
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → -𝑒𝑧 ∈ ℝ*) |
| 23 | | simpr 110 |
. . . . . . 7
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → 𝑦 ∈ ℝ*) |
| 24 | 23 | xnegcld 9930 |
. . . . . 6
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → -𝑒𝑦 ∈ ℝ*) |
| 25 | | brcnvg 4847 |
. . . . . 6
⊢
((-𝑒𝑧 ∈ ℝ* ∧
-𝑒𝑦
∈ ℝ*) → (-𝑒𝑧◡
< -𝑒𝑦
↔ -𝑒𝑦 < -𝑒𝑧)) |
| 26 | 22, 24, 25 | syl2anc 411 |
. . . . 5
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (-𝑒𝑧◡
< -𝑒𝑦
↔ -𝑒𝑦 < -𝑒𝑧)) |
| 27 | | xnegeq 9902 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → -𝑒𝑥 = -𝑒𝑧) |
| 28 | 1, 27, 21, 22 | fvmptd3 5655 |
. . . . . 6
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝐹‘𝑧) = -𝑒𝑧) |
| 29 | | xnegeq 9902 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → -𝑒𝑥 = -𝑒𝑦) |
| 30 | 1, 29, 23, 24 | fvmptd3 5655 |
. . . . . 6
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝐹‘𝑦) = -𝑒𝑦) |
| 31 | 28, 30 | breq12d 4046 |
. . . . 5
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → ((𝐹‘𝑧)◡
< (𝐹‘𝑦) ↔
-𝑒𝑧◡ < -𝑒𝑦)) |
| 32 | | xltneg 9911 |
. . . . 5
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑧 < 𝑦 ↔ -𝑒𝑦 < -𝑒𝑧)) |
| 33 | 26, 31, 32 | 3bitr4rd 221 |
. . . 4
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡
< (𝐹‘𝑦))) |
| 34 | 33 | rgen2a 2551 |
. . 3
⊢
∀𝑧 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡
< (𝐹‘𝑦)) |
| 35 | | df-isom 5267 |
. . 3
⊢ (𝐹 Isom < , ◡ < (ℝ*,
ℝ*) ↔ (𝐹:ℝ*–1-1-onto→ℝ* ∧ ∀𝑧 ∈ ℝ*
∀𝑦 ∈
ℝ* (𝑧 <
𝑦 ↔ (𝐹‘𝑧)◡
< (𝐹‘𝑦)))) |
| 36 | 20, 34, 35 | mpbir2an 944 |
. 2
⊢ 𝐹 Isom < , ◡ < (ℝ*,
ℝ*) |
| 37 | | xnegeq 9902 |
. . . 4
⊢ (𝑦 = 𝑥 → -𝑒𝑦 = -𝑒𝑥) |
| 38 | 37 | cbvmptv 4129 |
. . 3
⊢ (𝑦 ∈ ℝ*
↦ -𝑒𝑦) = (𝑥 ∈ ℝ* ↦
-𝑒𝑥) |
| 39 | 19 | simpri 113 |
. . 3
⊢ ◡𝐹 = (𝑦 ∈ ℝ* ↦
-𝑒𝑦) |
| 40 | 38, 39, 1 | 3eqtr4i 2227 |
. 2
⊢ ◡𝐹 = 𝐹 |
| 41 | 36, 40 | pm3.2i 272 |
1
⊢ (𝐹 Isom < , ◡ < (ℝ*,
ℝ*) ∧ ◡𝐹 = 𝐹) |