Step | Hyp | Ref
| Expression |
1 | | simp1 997 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐹 Isom < , < (𝐴, 𝐵)) |
2 | | simp3r 1026 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴) |
3 | | simp3l 1025 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴) |
4 | | isorel 5811 |
. . . 4
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐷 < 𝐶 ↔ (𝐹‘𝐷) < (𝐹‘𝐶))) |
5 | 4 | notbid 667 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶))) |
6 | 1, 2, 3, 5 | syl12anc 1236 |
. 2
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶))) |
7 | | simp2l 1023 |
. . . 4
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐴 ⊆
ℝ*) |
8 | 7, 3 | sseldd 3158 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈
ℝ*) |
9 | 7, 2 | sseldd 3158 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈
ℝ*) |
10 | | xrlenlt 8024 |
. . 3
⊢ ((𝐶 ∈ ℝ*
∧ 𝐷 ∈
ℝ*) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶)) |
11 | 8, 9, 10 | syl2anc 411 |
. 2
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶)) |
12 | | simp2r 1024 |
. . . 4
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ⊆
ℝ*) |
13 | | isof1o 5810 |
. . . . . 6
⊢ (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) |
14 | | f1of 5463 |
. . . . . 6
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) |
15 | 1, 13, 14 | 3syl 17 |
. . . . 5
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵) |
16 | 15, 3 | ffvelcdmd 5654 |
. . . 4
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ 𝐵) |
17 | 12, 16 | sseldd 3158 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈
ℝ*) |
18 | 15, 2 | ffvelcdmd 5654 |
. . . 4
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ 𝐵) |
19 | 12, 18 | sseldd 3158 |
. . 3
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈
ℝ*) |
20 | | xrlenlt 8024 |
. . 3
⊢ (((𝐹‘𝐶) ∈ ℝ* ∧ (𝐹‘𝐷) ∈ ℝ*) → ((𝐹‘𝐶) ≤ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶))) |
21 | 17, 19, 20 | syl2anc 411 |
. 2
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) ≤ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶))) |
22 | 6, 11, 21 | 3bitr4d 220 |
1
⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*)
∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))) |