Step |
Hyp |
Ref |
Expression |
1 |
|
strlem3a.1 |
⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ) |
2 |
|
id |
⊢ ( 𝑥 ∈ Cℋ → 𝑥 ∈ Cℋ ) |
3 |
|
simpl |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑢 ∈ ℋ ) |
4 |
|
pjhcl |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ ) |
5 |
2 3 4
|
syl2anr |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ ) |
6 |
|
normcl |
⊢ ( ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ) |
7 |
5 6
|
syl |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ) |
8 |
7
|
resqcld |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ℝ ) |
9 |
7
|
sqge0d |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → 0 ≤ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ) |
10 |
|
normge0 |
⊢ ( ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ∈ ℋ → 0 ≤ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ) |
11 |
5 10
|
syl |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → 0 ≤ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ) |
12 |
|
pjnorm |
⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ≤ ( normℎ ‘ 𝑢 ) ) |
13 |
2 3 12
|
syl2anr |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ≤ ( normℎ ‘ 𝑢 ) ) |
14 |
|
simplr |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( normℎ ‘ 𝑢 ) = 1 ) |
15 |
13 14
|
breqtrd |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ≤ 1 ) |
16 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
17 |
|
exple1 |
⊢ ( ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∧ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ≤ 1 ) ∧ 2 ∈ ℕ0 ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 ) |
18 |
16 17
|
mpan2 |
⊢ ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ∧ ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ≤ 1 ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 ) |
19 |
7 11 15 18
|
syl3anc |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 ) |
20 |
|
elicc01 |
⊢ ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∧ ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ≤ 1 ) ) |
21 |
8 9 19 20
|
syl3anbrc |
⊢ ( ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) ∧ 𝑥 ∈ Cℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) ↑ 2 ) ∈ ( 0 [,] 1 ) ) |
22 |
21 1
|
fmptd |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ) |
23 |
|
helch |
⊢ ℋ ∈ Cℋ |
24 |
1
|
strlem2 |
⊢ ( ℋ ∈ Cℋ → ( 𝑆 ‘ ℋ ) = ( ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) ) |
25 |
23 24
|
ax-mp |
⊢ ( 𝑆 ‘ ℋ ) = ( ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) |
26 |
|
pjch1 |
⊢ ( 𝑢 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) = 𝑢 ) |
27 |
26
|
fveq2d |
⊢ ( 𝑢 ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) = ( normℎ ‘ 𝑢 ) ) |
28 |
27
|
oveq1d |
⊢ ( 𝑢 ∈ ℋ → ( ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( normℎ ‘ 𝑢 ) ↑ 2 ) ) |
29 |
|
oveq1 |
⊢ ( ( normℎ ‘ 𝑢 ) = 1 → ( ( normℎ ‘ 𝑢 ) ↑ 2 ) = ( 1 ↑ 2 ) ) |
30 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
31 |
29 30
|
eqtrdi |
⊢ ( ( normℎ ‘ 𝑢 ) = 1 → ( ( normℎ ‘ 𝑢 ) ↑ 2 ) = 1 ) |
32 |
28 31
|
sylan9eq |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( ( normℎ ‘ ( ( projℎ ‘ ℋ ) ‘ 𝑢 ) ) ↑ 2 ) = 1 ) |
33 |
25 32
|
syl5eq |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( 𝑆 ‘ ℋ ) = 1 ) |
34 |
|
pjcjt2 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ) |
35 |
34
|
imp |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) = ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) |
36 |
35
|
fveq2d |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( normℎ ‘ ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) = ( normℎ ‘ ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ) |
37 |
36
|
oveq1d |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( normℎ ‘ ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( normℎ ‘ ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) ) |
38 |
|
pjopyth |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( ( normℎ ‘ ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) ) |
39 |
38
|
imp |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( normℎ ‘ ( ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) +ℎ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) |
40 |
37 39
|
eqtrd |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( normℎ ‘ ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) |
41 |
|
chjcl |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ ) |
42 |
41
|
3adant3 |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ ) |
44 |
1
|
strlem2 |
⊢ ( ( 𝑧 ∨ℋ 𝑤 ) ∈ Cℋ → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( normℎ ‘ ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) ) |
45 |
43 44
|
syl |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( normℎ ‘ ( ( projℎ ‘ ( 𝑧 ∨ℋ 𝑤 ) ) ‘ 𝑢 ) ) ↑ 2 ) ) |
46 |
|
3simpa |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) → ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) ) |
47 |
46
|
adantr |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) ) |
48 |
1
|
strlem2 |
⊢ ( 𝑧 ∈ Cℋ → ( 𝑆 ‘ 𝑧 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) ) |
49 |
1
|
strlem2 |
⊢ ( 𝑤 ∈ Cℋ → ( 𝑆 ‘ 𝑤 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) |
50 |
48 49
|
oveqan12d |
⊢ ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ) → ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) = ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) |
51 |
47 50
|
syl |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) = ( ( ( normℎ ‘ ( ( projℎ ‘ 𝑧 ) ‘ 𝑢 ) ) ↑ 2 ) + ( ( normℎ ‘ ( ( projℎ ‘ 𝑤 ) ‘ 𝑢 ) ) ↑ 2 ) ) ) |
52 |
40 45 51
|
3eqtr4d |
⊢ ( ( ( 𝑧 ∈ Cℋ ∧ 𝑤 ∈ Cℋ ∧ 𝑢 ∈ ℋ ) ∧ 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) |
53 |
52
|
3exp1 |
⊢ ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑢 ∈ ℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) |
54 |
53
|
com3r |
⊢ ( 𝑢 ∈ ℋ → ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) |
55 |
54
|
adantr |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( 𝑧 ∈ Cℋ → ( 𝑤 ∈ Cℋ → ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) ) ) |
56 |
55
|
ralrimdv |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( 𝑧 ∈ Cℋ → ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) ) |
57 |
56
|
ralrimiv |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ∀ 𝑧 ∈ Cℋ ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) |
58 |
|
isst |
⊢ ( 𝑆 ∈ States ↔ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑧 ∈ Cℋ ∀ 𝑤 ∈ Cℋ ( 𝑧 ⊆ ( ⊥ ‘ 𝑤 ) → ( 𝑆 ‘ ( 𝑧 ∨ℋ 𝑤 ) ) = ( ( 𝑆 ‘ 𝑧 ) + ( 𝑆 ‘ 𝑤 ) ) ) ) ) |
59 |
22 33 57 58
|
syl3anbrc |
⊢ ( ( 𝑢 ∈ ℋ ∧ ( normℎ ‘ 𝑢 ) = 1 ) → 𝑆 ∈ States ) |