Step |
Hyp |
Ref |
Expression |
1 |
|
pjop.1 |
⊢ 𝐻 ∈ Cℋ |
2 |
|
pjop.2 |
⊢ 𝐴 ∈ ℋ |
3 |
1 2
|
pjhclii |
⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ |
4 |
|
ax-hvaddid |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) |
5 |
3 4
|
ax-mp |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) |
6 |
1 2
|
pjpji |
⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) |
7 |
1 2
|
pjoc1i |
⊢ ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ) |
8 |
7
|
biimpi |
⊢ ( 𝐴 ∈ 𝐻 → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ) |
9 |
8
|
oveq2d |
⊢ ( 𝐴 ∈ 𝐻 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) ) |
10 |
6 9
|
eqtr2id |
⊢ ( 𝐴 ∈ 𝐻 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = 𝐴 ) |
11 |
5 10
|
eqtr3id |
⊢ ( 𝐴 ∈ 𝐻 → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) |
12 |
1 2
|
pjclii |
⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 |
13 |
|
eleq1 |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 ↔ 𝐴 ∈ 𝐻 ) ) |
14 |
12 13
|
mpbii |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 → 𝐴 ∈ 𝐻 ) |
15 |
11 14
|
impbii |
⊢ ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) |