Step |
Hyp |
Ref |
Expression |
1 |
|
hstrlem3a.1 |
⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) |
2 |
|
pjhcl |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ ) |
3 |
2
|
ancoms |
⊢ ( ( 𝑢 ∈ ℋ ∧ 𝑥 ∈ Cℋ ) → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ ) |
4 |
3
|
adantlr |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ ) |
5 |
4 1
|
fmptd |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 : Cℋ ⟶ ℋ ) |
6 |
|
helch |
⊢ ℋ ∈ Cℋ |
7 |
1
|
hstrlem2 |
⊢ ( ℋ ∈ Cℋ → ( 𝑆 ‘ ℋ ) = ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) |
8 |
6 7
|
ax-mp |
⊢ ( 𝑆 ‘ ℋ ) = ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) |
9 |
8
|
fveq2i |
⊢ ( normℎ ‘ ( 𝑆 ‘ ℋ ) ) = ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) |
10 |
|
pjch1 |
⊢ ( 𝑢 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) = 𝑢 ) |
11 |
10
|
fveq2d |
⊢ ( 𝑢 ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) = ( normℎ ‘ 𝑢 ) ) |
12 |
|
id |
⊢ ( ( normℎ ‘ 𝑢 ) = 1 → ( normℎ ‘ 𝑢 ) = 1 ) |
13 |
11 12
|
sylan9eq |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) = 1 ) |
14 |
9 13
|
syl5eq |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( normℎ ‘ ( 𝑆 ‘ ℋ ) ) = 1 ) |
15 |
1
|
hstrlem2 |
⊢ ( 𝑧 ∈ Cℋ → ( 𝑆 ‘ 𝑧 ) = ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) |
16 |
1
|
hstrlem2 |
⊢ ( 𝑤 ∈ Cℋ → ( 𝑆 ‘ 𝑤 ) = ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) |
17 |
15 16
|
oveqan12d |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ·ih ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ·ih ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ·ih ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
20 |
|
pjoi0 |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ·ih ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) = 0 ) |
21 |
19 20
|
eqtrd |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ) |
22 |
|
pjcjt2 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ) |
23 |
22
|
imp |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
24 |
|
chjcl |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ ) |
25 |
1
|
hstrlem2 |
⊢ ( ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) |
26 |
24 25
|
syl |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) |
27 |
26
|
3adant3 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) |
28 |
27
|
adantr |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) |
29 |
15 16
|
oveqan12d |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
30 |
29
|
3adant3 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
32 |
23 28 31
|
3eqtr4d |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) |
33 |
21 32
|
jca |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) |
34 |
33
|
3exp1 |
⊢ ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑢 ∈ ℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) ) |
35 |
34
|
com3r |
⊢ ( 𝑢 ∈ ℋ → ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) ) |
36 |
35
|
adantr |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) ) |
37 |
36
|
ralrimdv |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( 𝑧 ∈ Cℋ → ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) |
38 |
37
|
ralrimiv |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ∀ 𝑧 ∈ Cℋ ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) |
39 |
|
ishst |
⊢ ( 𝑆 ∈ CHStates ↔ ( 𝑆 : Cℋ ⟶ ℋ ∧ ( normℎ ‘ ( 𝑆 ‘ ℋ ) ) = 1 ∧ ∀ 𝑧 ∈ Cℋ ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( ( 𝑆 ‘ 𝑧 ) ·ih ( 𝑆 ‘ 𝑤 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) +ℎ ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) |
40 |
5 14 38 39
|
syl3anbrc |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ CHStates ) |