Step | Hyp | Ref
| Expression |
1 | | elunop 30213 |
. . . . 5
⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) |
2 | 1 | simplbi 497 |
. . . 4
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ) |
3 | | fof 6684 |
. . . 4
⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ) |
4 | 2, 3 | syl 17 |
. . 3
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶
ℋ) |
5 | | unop 30256 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) |
6 | 5 | 3anidm23 1419 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) |
7 | 6 | 3adant3 1130 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) |
8 | | unop 30256 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) |
9 | 8 | 3anidm23 1419 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) |
10 | 9 | 3adant2 1129 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) |
11 | 7, 10 | oveq12d 7286 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘𝑥) ·ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑦))) = ((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦))) |
12 | | unop 30256 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
13 | | unop 30256 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑥)) = (𝑦 ·ih 𝑥)) |
14 | 13 | 3com23 1124 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑥)) = (𝑦 ·ih 𝑥)) |
15 | 12, 14 | oveq12d 7286 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑥))) = ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥))) |
16 | 11, 15 | oveq12d 7286 |
. . . . . . . . 9
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
((((𝑇‘𝑥)
·ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑥)))) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥)))) |
17 | 16 | 3expb 1118 |
. . . . . . . 8
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
((((𝑇‘𝑥)
·ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑥)))) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥)))) |
18 | | ffvelrn 6953 |
. . . . . . . . . . 11
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
(𝑇‘𝑥) ∈ ℋ) |
19 | | ffvelrn 6953 |
. . . . . . . . . . 11
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑦 ∈ ℋ) →
(𝑇‘𝑦) ∈ ℋ) |
20 | 18, 19 | anim12dan 618 |
. . . . . . . . . 10
⊢ ((𝑇: ℋ⟶ ℋ ∧
(𝑥 ∈ ℋ ∧
𝑦 ∈ ℋ)) →
((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) |
21 | 4, 20 | sylan 579 |
. . . . . . . . 9
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) |
22 | | normlem9at 29462 |
. . . . . . . . 9
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = ((((𝑇‘𝑥) ·ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑥))))) |
23 | 21, 22 | syl 17 |
. . . . . . . 8
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
(((𝑇‘𝑥) −ℎ
(𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = ((((𝑇‘𝑥) ·ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·ih (𝑇‘𝑥))))) |
24 | | normlem9at 29462 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −ℎ
𝑦)
·ih (𝑥 −ℎ 𝑦)) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥)))) |
25 | 24 | adantl 481 |
. . . . . . . 8
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 −ℎ
𝑦)
·ih (𝑥 −ℎ 𝑦)) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥)))) |
26 | 17, 23, 25 | 3eqtr4rd 2790 |
. . . . . . 7
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 −ℎ
𝑦)
·ih (𝑥 −ℎ 𝑦)) = (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦)))) |
27 | 26 | eqeq1d 2741 |
. . . . . 6
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
(((𝑥
−ℎ 𝑦) ·ih (𝑥 −ℎ
𝑦)) = 0 ↔ (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = 0)) |
28 | | hvsubcl 29358 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −ℎ
𝑦) ∈
ℋ) |
29 | | his6 29440 |
. . . . . . . . 9
⊢ ((𝑥 −ℎ
𝑦) ∈ ℋ →
(((𝑥
−ℎ 𝑦) ·ih (𝑥 −ℎ
𝑦)) = 0 ↔ (𝑥 −ℎ
𝑦) =
0ℎ)) |
30 | 28, 29 | syl 17 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 −ℎ
𝑦)
·ih (𝑥 −ℎ 𝑦)) = 0 ↔ (𝑥 −ℎ
𝑦) =
0ℎ)) |
31 | | hvsubeq0 29409 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −ℎ
𝑦) = 0ℎ
↔ 𝑥 = 𝑦)) |
32 | 30, 31 | bitrd 278 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 −ℎ
𝑦)
·ih (𝑥 −ℎ 𝑦)) = 0 ↔ 𝑥 = 𝑦)) |
33 | 32 | adantl 481 |
. . . . . 6
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
(((𝑥
−ℎ 𝑦) ·ih (𝑥 −ℎ
𝑦)) = 0 ↔ 𝑥 = 𝑦)) |
34 | | hvsubcl 29358 |
. . . . . . . . 9
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ∈ ℋ) |
35 | | his6 29440 |
. . . . . . . . 9
⊢ (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ∈ ℋ → ((((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = 0 ↔ ((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) = 0ℎ)) |
36 | 34, 35 | syl 17 |
. . . . . . . 8
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = 0 ↔ ((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) = 0ℎ)) |
37 | | hvsubeq0 29409 |
. . . . . . . 8
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) = 0ℎ ↔ (𝑇‘𝑥) = (𝑇‘𝑦))) |
38 | 36, 37 | bitrd 278 |
. . . . . . 7
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = 0 ↔ (𝑇‘𝑥) = (𝑇‘𝑦))) |
39 | 21, 38 | syl 17 |
. . . . . 6
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
((((𝑇‘𝑥) −ℎ
(𝑇‘𝑦)) ·ih
((𝑇‘𝑥) −ℎ
(𝑇‘𝑦))) = 0 ↔ (𝑇‘𝑥) = (𝑇‘𝑦))) |
40 | 27, 33, 39 | 3bitr3rd 309 |
. . . . 5
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) ↔ 𝑥 = 𝑦)) |
41 | 40 | biimpd 228 |
. . . 4
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)) |
42 | 41 | ralrimivva 3116 |
. . 3
⊢ (𝑇 ∈ UniOp →
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)) |
43 | | dff13 7122 |
. . 3
⊢ (𝑇: ℋ–1-1→ ℋ ↔ (𝑇: ℋ⟶ ℋ ∧
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))) |
44 | 4, 42, 43 | sylanbrc 582 |
. 2
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1→ ℋ) |
45 | | df-f1o 6437 |
. 2
⊢ (𝑇: ℋ–1-1-onto→
ℋ ↔ (𝑇:
ℋ–1-1→ ℋ ∧
𝑇: ℋ–onto→ ℋ)) |
46 | 44, 2, 45 | sylanbrc 582 |
1
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→
ℋ) |