Step |
Hyp |
Ref |
Expression |
1 |
|
hstrlem3.1 |
⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) ) |
2 |
|
hstrlem3.2 |
⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) ) |
3 |
|
hstrlem3.3 |
⊢ 𝐴 ∈ Cℋ |
4 |
|
hstrlem3.4 |
⊢ 𝐵 ∈ Cℋ |
5 |
1 2 3 4
|
hstrlem4 |
⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 ) |
6 |
1 2 3 4
|
hstrlem3 |
⊢ ( 𝜑 → 𝑆 ∈ CHStates ) |
7 |
|
hstcl |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) |
8 |
6 4 7
|
sylancl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) |
9 |
|
normcl |
⊢ ( ( 𝑆 ‘ 𝐵 ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ) |
11 |
1 2 3 4
|
hstrlem5 |
⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) < 1 ) |
12 |
10 11
|
ltned |
⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≠ 1 ) |
13 |
12
|
neneqd |
⊢ ( 𝜑 → ¬ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ) |
14 |
5 13
|
jcnd |
⊢ ( 𝜑 → ¬ ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ) ) |