Step |
Hyp |
Ref |
Expression |
1 |
|
tendo0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tendo0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
tendo0.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendo0.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
tendo0.o |
⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
6 |
2 3
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ) |
7 |
5 1
|
tendo02 |
⊢ ( ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 → ( 𝑂 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( I ↾ 𝐵 ) ) |
8 |
6 7
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑂 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( I ↾ 𝐵 ) ) |
9 |
5 1
|
tendo02 |
⊢ ( 𝐹 ∈ 𝑇 → ( 𝑂 ‘ 𝐹 ) = ( I ↾ 𝐵 ) ) |
10 |
9
|
3ad2ant2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑂 ‘ 𝐹 ) = ( I ↾ 𝐵 ) ) |
11 |
5 1
|
tendo02 |
⊢ ( 𝐺 ∈ 𝑇 → ( 𝑂 ‘ 𝐺 ) = ( I ↾ 𝐵 ) ) |
12 |
11
|
3ad2ant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑂 ‘ 𝐺 ) = ( I ↾ 𝐵 ) ) |
13 |
10 12
|
coeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑂 ‘ 𝐹 ) ∘ ( 𝑂 ‘ 𝐺 ) ) = ( ( I ↾ 𝐵 ) ∘ ( I ↾ 𝐵 ) ) ) |
14 |
|
f1oi |
⊢ ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵 |
15 |
|
f1of |
⊢ ( ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵 → ( I ↾ 𝐵 ) : 𝐵 ⟶ 𝐵 ) |
16 |
|
fcoi1 |
⊢ ( ( I ↾ 𝐵 ) : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ) |
17 |
14 15 16
|
mp2b |
⊢ ( ( I ↾ 𝐵 ) ∘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) |
18 |
13 17
|
eqtr2di |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( I ↾ 𝐵 ) = ( ( 𝑂 ‘ 𝐹 ) ∘ ( 𝑂 ‘ 𝐺 ) ) ) |
19 |
8 18
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑂 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑂 ‘ 𝐹 ) ∘ ( 𝑂 ‘ 𝐺 ) ) ) |