Step |
Hyp |
Ref |
Expression |
1 |
|
pjrn |
⊢ ( 𝐺 ∈ Cℋ → ran ( projℎ ‘ 𝐺 ) = 𝐺 ) |
2 |
1
|
adantr |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ) → ran ( projℎ ‘ 𝐺 ) = 𝐺 ) |
3 |
|
pjrn |
⊢ ( 𝐻 ∈ Cℋ → ran ( projℎ ‘ 𝐻 ) = 𝐻 ) |
4 |
3
|
fveq2d |
⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) = ( ⊥ ‘ 𝐻 ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ) → ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) = ( ⊥ ‘ 𝐻 ) ) |
6 |
2 5
|
sseq12d |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ) → ( ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ↔ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) ) |
7 |
6
|
biimpar |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) |
8 |
7
|
3adantl3 |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) |
9 |
|
id |
⊢ ( 𝐻 ∈ Cℋ → 𝐻 ∈ Cℋ ) |
10 |
3 9
|
eqeltrd |
⊢ ( 𝐻 ∈ Cℋ → ran ( projℎ ‘ 𝐻 ) ∈ Cℋ ) |
11 |
|
chsh |
⊢ ( ran ( projℎ ‘ 𝐻 ) ∈ Cℋ → ran ( projℎ ‘ 𝐻 ) ∈ Sℋ ) |
12 |
10 11
|
syl |
⊢ ( 𝐻 ∈ Cℋ → ran ( projℎ ‘ 𝐻 ) ∈ Sℋ ) |
13 |
12
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ran ( projℎ ‘ 𝐻 ) ∈ Sℋ ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) → ran ( projℎ ‘ 𝐻 ) ∈ Sℋ ) |
15 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) → ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) |
16 |
|
pjfn |
⊢ ( 𝐺 ∈ Cℋ → ( projℎ ‘ 𝐺 ) Fn ℋ ) |
17 |
|
fnfvelrn |
⊢ ( ( ( projℎ ‘ 𝐺 ) Fn ℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ) |
18 |
16 17
|
sylan |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ) |
19 |
18
|
3adant2 |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ) |
20 |
|
pjfn |
⊢ ( 𝐻 ∈ Cℋ → ( projℎ ‘ 𝐻 ) Fn ℋ ) |
21 |
|
fnfvelrn |
⊢ ( ( ( projℎ ‘ 𝐻 ) Fn ℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) |
22 |
20 21
|
sylan |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) |
23 |
22
|
3adant1 |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) |
24 |
19 23
|
jca |
⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) ) |
26 |
|
shorth |
⊢ ( ran ( projℎ ‘ 𝐻 ) ∈ Sℋ → ( ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) → ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐺 ) ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ran ( projℎ ‘ 𝐻 ) ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ·ih ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 0 ) ) ) |
27 |
14 15 25 26
|
syl3c |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ ran ( projℎ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ran ( projℎ ‘ 𝐻 ) ) ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ·ih ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 0 ) |
28 |
8 27
|
syldan |
⊢ ( ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ·ih ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 0 ) |