Step |
Hyp |
Ref |
Expression |
1 |
|
dihcnvid1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihcnvid1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dihcnvid1.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
eqid |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
eqid |
⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
6 |
1 2 3 4 5
|
dihf11 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
7 |
|
f1f1orn |
⊢ ( 𝐼 : 𝐵 –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) → 𝐼 : 𝐵 –1-1-onto→ ran 𝐼 ) |
8 |
6 7
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 –1-1-onto→ ran 𝐼 ) |
9 |
|
f1ocnvfv1 |
⊢ ( ( 𝐼 : 𝐵 –1-1-onto→ ran 𝐼 ∧ 𝑋 ∈ 𝐵 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
10 |
8 9
|
sylan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |