Step |
Hyp |
Ref |
Expression |
1 |
|
pjidm.1 |
⊢ 𝐻 ∈ Cℋ |
2 |
|
pjidm.2 |
⊢ 𝐴 ∈ ℋ |
3 |
|
pjsslem.1 |
⊢ 𝐺 ∈ Cℋ |
4 |
1
|
choccli |
⊢ ( ⊥ ‘ 𝐻 ) ∈ Cℋ |
5 |
3 4
|
chincli |
⊢ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ∈ Cℋ |
6 |
3 2
|
pjhclii |
⊢ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ℋ |
7 |
1 2
|
pjhclii |
⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ |
8 |
5 6 7
|
pjsubii |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) −ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
9 |
5 6
|
pjhclii |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) ∈ ℋ |
10 |
5 7
|
pjhclii |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℋ |
11 |
9 10
|
hvsubvali |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) −ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) +ℎ ( - 1 ·ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) ) |
12 |
|
inss1 |
⊢ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ⊆ 𝐺 |
13 |
5 2 3
|
pjss2i |
⊢ ( ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ⊆ 𝐺 → ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) |
14 |
12 13
|
ax-mp |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) |
15 |
1
|
chshii |
⊢ 𝐻 ∈ Sℋ |
16 |
|
shococss |
⊢ ( 𝐻 ∈ Sℋ → 𝐻 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) ) |
17 |
15 16
|
ax-mp |
⊢ 𝐻 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) |
18 |
|
inss2 |
⊢ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ⊆ ( ⊥ ‘ 𝐻 ) |
19 |
5 4
|
chsscon3i |
⊢ ( ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ⊆ ( ⊥ ‘ 𝐻 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ) |
20 |
18 19
|
mpbi |
⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) |
21 |
17 20
|
sstri |
⊢ 𝐻 ⊆ ( ⊥ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) |
22 |
1 2
|
pjclii |
⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 |
23 |
21 22
|
sselii |
⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) |
24 |
5 7
|
pjoc2i |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ↔ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 0ℎ ) |
25 |
23 24
|
mpbi |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 0ℎ |
26 |
25
|
oveq2i |
⊢ ( - 1 ·ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( - 1 ·ℎ 0ℎ ) |
27 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
28 |
|
hvmul0 |
⊢ ( - 1 ∈ ℂ → ( - 1 ·ℎ 0ℎ ) = 0ℎ ) |
29 |
27 28
|
ax-mp |
⊢ ( - 1 ·ℎ 0ℎ ) = 0ℎ |
30 |
26 29
|
eqtri |
⊢ ( - 1 ·ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = 0ℎ |
31 |
14 30
|
oveq12i |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) +ℎ ( - 1 ·ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) ) = ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) +ℎ 0ℎ ) |
32 |
5 2
|
pjhclii |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ∈ ℋ |
33 |
|
ax-hvaddid |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ∈ ℋ → ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) |
34 |
32 33
|
ax-mp |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) |
35 |
31 34
|
eqtri |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) +ℎ ( - 1 ·ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) |
36 |
11 35
|
eqtri |
⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) −ℎ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) |
37 |
8 36
|
eqtri |
⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) |
38 |
3 2
|
pjclii |
⊢ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝐺 |
39 |
|
ssel |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐺 ) ) |
40 |
22 39
|
mpi |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐺 ) |
41 |
3
|
chshii |
⊢ 𝐺 ∈ Sℋ |
42 |
|
shsubcl |
⊢ ( ( 𝐺 ∈ Sℋ ∧ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝐺 ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐺 ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ) |
43 |
41 42
|
mp3an1 |
⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ 𝐺 ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐺 ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ) |
44 |
38 40 43
|
sylancr |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ) |
45 |
1 2 3
|
pjsslem |
⊢ ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) |
46 |
1 3
|
chsscon3i |
⊢ ( 𝐻 ⊆ 𝐺 ↔ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ 𝐻 ) ) |
47 |
4 2
|
pjclii |
⊢ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) |
48 |
3
|
choccli |
⊢ ( ⊥ ‘ 𝐺 ) ∈ Cℋ |
49 |
48 2
|
pjclii |
⊢ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) |
50 |
|
ssel |
⊢ ( ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ 𝐻 ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ) ) |
51 |
49 50
|
mpi |
⊢ ( ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ) |
52 |
4
|
chshii |
⊢ ( ⊥ ‘ 𝐻 ) ∈ Sℋ |
53 |
|
shsubcl |
⊢ ( ( ( ⊥ ‘ 𝐻 ) ∈ Sℋ ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) |
54 |
52 53
|
mp3an1 |
⊢ ( ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐻 ) ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) |
55 |
47 51 54
|
sylancr |
⊢ ( ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ 𝐻 ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) |
56 |
46 55
|
sylbi |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) |
57 |
45 56
|
eqeltrrid |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) |
58 |
44 57
|
jca |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ∧ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) ) |
59 |
|
elin |
⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ↔ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ∧ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) ) |
60 |
6 7
|
hvsubcli |
⊢ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℋ |
61 |
5 60
|
pjchi |
⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ↔ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
62 |
59 61
|
bitr3i |
⊢ ( ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ 𝐺 ∧ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ 𝐻 ) ) ↔ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
63 |
58 62
|
sylib |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) |
64 |
37 63
|
syl5reqr |
⊢ ( 𝐻 ⊆ 𝐺 → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) |