| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elunop 31891 | . . . . 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 6820 | . . . 4
⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ) | 
| 4 | 2, 3 | syl 17 | . . 3
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶
ℋ) | 
| 5 |  | unop 31934 | . . . . . . . . . . . . 13
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) | 
| 6 | 5 | 3anidm23 1423 | . . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) | 
| 7 | 6 | 3adant3 1133 | . . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (𝑥 ·ih 𝑥)) | 
| 8 |  | unop 31934 | . . . . . . . . . . . . 13
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) | 
| 9 | 8 | 3anidm23 1423 | . . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) | 
| 10 | 9 | 3adant2 1132 | . . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑦)) = (𝑦 ·ih 𝑦)) | 
| 11 | 7, 10 | oveq12d 7449 | . . . . . . . . . 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 31934 | . . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) | 
| 13 |  | unop 31934 | . . . . . . . . . . . 12
⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑥)) = (𝑦 ·ih 𝑥)) | 
| 14 | 13 | 3com23 1127 | . . . . . . . . . . 11
⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·ih (𝑇‘𝑥)) = (𝑦 ·ih 𝑥)) | 
| 15 | 12, 14 | oveq12d 7449 | . . . . . . . . . 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 7449 | . . . . . . . . 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 1121 | . . . . . . . 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 |  | ffvelcdm 7101 | . . . . . . . . . . 11
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
(𝑇‘𝑥) ∈ ℋ) | 
| 19 |  | ffvelcdm 7101 | . . . . . . . . . . 11
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑦 ∈ ℋ) →
(𝑇‘𝑦) ∈ ℋ) | 
| 20 | 18, 19 | anim12dan 619 | . . . . . . . . . 10
⊢ ((𝑇: ℋ⟶ ℋ ∧
(𝑥 ∈ ℋ ∧
𝑦 ∈ ℋ)) →
((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) | 
| 21 | 4, 20 | sylan 580 | . . . . . . . . 9
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) | 
| 22 |  | normlem9at 31140 | . . . . . . . . 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 31140 | . . . . . . . . 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 2788 | . . . . . . 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 2739 | . . . . . 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 31036 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −ℎ
𝑦) ∈
ℋ) | 
| 29 |  | his6 31118 | . . . . . . . . 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 31087 | . . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −ℎ
𝑦) = 0ℎ
↔ 𝑥 = 𝑦)) | 
| 32 | 30, 31 | bitrd 279 | . . . . . . 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 31036 | . . . . . . . . 9
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) ∈ ℋ) | 
| 35 |  | his6 31118 | . . . . . . . . 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 31087 | . . . . . . . 8
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −ℎ (𝑇‘𝑦)) = 0ℎ ↔ (𝑇‘𝑥) = (𝑇‘𝑦))) | 
| 38 | 36, 37 | bitrd 279 | . . . . . . 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 310 | . . . . 5
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) ↔ 𝑥 = 𝑦)) | 
| 41 | 40 | biimpd 229 | . . . 4
⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)) | 
| 42 | 41 | ralrimivva 3202 | . . 3
⊢ (𝑇 ∈ UniOp →
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)) | 
| 43 |  | dff13 7275 | . . 3
⊢ (𝑇: ℋ–1-1→ ℋ ↔ (𝑇: ℋ⟶ ℋ ∧
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))) | 
| 44 | 4, 42, 43 | sylanbrc 583 | . 2
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1→ ℋ) | 
| 45 |  | df-f1o 6568 | . 2
⊢ (𝑇: ℋ–1-1-onto→
ℋ ↔ (𝑇:
ℋ–1-1→ ℋ ∧
𝑇: ℋ–onto→ ℋ)) | 
| 46 | 44, 2, 45 | sylanbrc 583 | 1
⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→
ℋ) |