Step |
Hyp |
Ref |
Expression |
1 |
|
tendocan.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tendocan.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendocan.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendocan.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝐾 ∈ HL ) |
6 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑊 ∈ 𝐻 ) |
7 |
|
simp21 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑈 ∈ 𝐸 ) |
8 |
|
simp22 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑉 ∈ 𝐸 ) |
9 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ) |
11 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → 𝐹 ∈ 𝑇 ) |
12 |
|
simp13r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) ) |
13 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ℎ ∈ 𝑇 ) |
14 |
11 12 13
|
3jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ∧ ℎ ∈ 𝑇 ) ) |
15 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ℎ ≠ ( I ↾ 𝐵 ) ) |
16 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
1 2 3 16 4
|
cdlemj3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ∧ ℎ ∈ 𝑇 ) ) ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) |
18 |
9 10 14 15 17
|
syl31anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) |
19 |
18
|
3exp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ( ℎ ∈ 𝑇 → ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) ) ) |
20 |
19
|
ralrimiv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ∀ ℎ ∈ 𝑇 ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) ) |
21 |
1 2 3 4
|
tendoeq2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ ℎ ∈ 𝑇 ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) ) → 𝑈 = 𝑉 ) |
22 |
5 6 7 8 20 21
|
syl221anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑈 = 𝑉 ) |