Step |
Hyp |
Ref |
Expression |
1 |
|
choc1 |
⊢ ( ⊥ ‘ ℋ ) = 0ℋ |
2 |
1
|
fveq2i |
⊢ ( projℎ ‘ ( ⊥ ‘ ℋ ) ) = ( projℎ ‘ 0ℋ ) |
3 |
|
df-h0op |
⊢ 0hop = ( projℎ ‘ 0ℋ ) |
4 |
2 3
|
eqtr4i |
⊢ ( projℎ ‘ ( ⊥ ‘ ℋ ) ) = 0hop |
5 |
4
|
fveq1i |
⊢ ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( 0hop ‘ 𝐴 ) |
6 |
|
helch |
⊢ ℋ ∈ Cℋ |
7 |
|
pjo |
⊢ ( ( ℋ ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) ) |
8 |
6 7
|
mpan |
⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) ) |
9 |
5 8
|
syl5eqr |
⊢ ( 𝐴 ∈ ℋ → ( 0hop ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) ) |
10 |
6
|
pjhcli |
⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ ) |
11 |
|
hvsubid |
⊢ ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) = 0ℎ ) |
12 |
10 11
|
syl |
⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) = 0ℎ ) |
13 |
9 12
|
eqtrd |
⊢ ( 𝐴 ∈ ℋ → ( 0hop ‘ 𝐴 ) = 0ℎ ) |