Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 −ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) |
2 |
1
|
oveq2d |
⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) ) |
3 |
|
oveq2 |
⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 −ℎ 𝐴 ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) |
4 |
2 3
|
eqeq12d |
⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐴 ) ↔ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) |
6 |
5
|
oveq2d |
⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ) |
7 |
|
oveq1 |
⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) |
8 |
6 7
|
eqeq12d |
⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) |
9 |
|
ifhvhv0 |
⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ |
10 |
|
ifhvhv0 |
⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ |
11 |
9 10
|
hvnegdii |
⊢ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) |
12 |
4 8 11
|
dedth2h |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐴 ) ) |