Step |
Hyp |
Ref |
Expression |
1 |
|
pjocin.1 |
⊢ 𝐺 ∈ Cℋ |
2 |
|
pjocin.2 |
⊢ 𝐻 ∈ Cℋ |
3 |
1 2
|
chincli |
⊢ ( 𝐺 ∩ 𝐻 ) ∈ Cℋ |
4 |
3
|
choccli |
⊢ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Cℋ |
5 |
4
|
cheli |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → 𝐴 ∈ ℋ ) |
6 |
|
pjpo |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ) |
7 |
1 5 6
|
sylancr |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ) |
8 |
|
inss1 |
⊢ ( 𝐺 ∩ 𝐻 ) ⊆ 𝐺 |
9 |
3 1
|
chsscon3i |
⊢ ( ( 𝐺 ∩ 𝐻 ) ⊆ 𝐺 ↔ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |
10 |
8 9
|
mpbi |
⊢ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) |
11 |
1
|
choccli |
⊢ ( ⊥ ‘ 𝐺 ) ∈ Cℋ |
12 |
11
|
pjcli |
⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) ) |
13 |
5 12
|
syl |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) ) |
14 |
10 13
|
sselid |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |
15 |
4
|
chshii |
⊢ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Sℋ |
16 |
|
shsubcl |
⊢ ( ( ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Sℋ ∧ 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |
17 |
15 16
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |
18 |
14 17
|
mpdan |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |
19 |
7 18
|
eqeltrd |
⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) |