Step |
Hyp |
Ref |
Expression |
1 |
|
pjsumt.1 |
⊢ 𝐺 ∈ Cℋ |
2 |
|
pjsumt.2 |
⊢ 𝐻 ∈ Cℋ |
3 |
1 2
|
osumi |
⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( 𝐺 +ℋ 𝐻 ) = ( 𝐺 ∨ℋ 𝐻 ) ) |
4 |
3
|
fveq2d |
⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) = ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ) |
5 |
4
|
fveq1d |
⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) ) |
6 |
1 2
|
chjcli |
⊢ ( 𝐺 ∨ℋ 𝐻 ) ∈ Cℋ |
7 |
|
pjid |
⊢ ( ( ( 𝐺 ∨ℋ 𝐻 ) ∈ Cℋ ∧ 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 ) |
8 |
6 7
|
mpan |
⊢ ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 ) |
9 |
5 8
|
sylan9eqr |
⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 ) |
10 |
6
|
cheli |
⊢ ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) → 𝐴 ∈ ℋ ) |
11 |
1 2
|
pjsumi |
⊢ ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) ) |
12 |
11
|
imp |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
13 |
10 12
|
sylan |
⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
14 |
9 13
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → 𝐴 = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |