Step |
Hyp |
Ref |
Expression |
1 |
|
hmoval.8 |
⊢ 𝐻 = ( HmOp ‘ 𝑈 ) |
2 |
|
hmoval.9 |
⊢ 𝐴 = ( 𝑈 adj 𝑈 ) |
3 |
1 2
|
hmoval |
⊢ ( 𝑈 ∈ NrmCVec → 𝐻 = { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } ) |
4 |
3
|
eleq2d |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝑇 ∈ 𝐻 ↔ 𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } ) ) |
5 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( 𝐴 ‘ 𝑡 ) = ( 𝐴 ‘ 𝑇 ) ) |
6 |
|
id |
⊢ ( 𝑡 = 𝑇 → 𝑡 = 𝑇 ) |
7 |
5 6
|
eqeq12d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝐴 ‘ 𝑡 ) = 𝑡 ↔ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) ) |
8 |
7
|
elrab |
⊢ ( 𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) ) |
9 |
4 8
|
syl6bb |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝑇 ∈ 𝐻 ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) ) ) |