| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simpr |
⊢ ( ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) ∧ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 ) |
| 2 |
|
hstle |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) |
| 3 |
2
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) ∧ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 ) → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) |
| 4 |
1 3
|
eqbrtrrd |
⊢ ( ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) ∧ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 ) → 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) |
| 5 |
4
|
ex |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 → 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) |
| 6 |
|
hstle1 |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ) |
| 7 |
6
|
ad2ant2r |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ) |
| 8 |
5 7
|
jctild |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ∧ 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
| 9 |
|
hstcl |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) |
| 10 |
|
normcl |
⊢ ( ( 𝑆 ‘ 𝐵 ) ∈ ℋ → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ) |
| 11 |
|
1re |
⊢ 1 ∈ ℝ |
| 12 |
|
letri3 |
⊢ ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ∧ 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
| 13 |
11 12
|
mpan2 |
⊢ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ ℝ → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ∧ 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
| 14 |
9 10 13
|
3syl |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ∧ 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
| 15 |
14
|
ad2ant2r |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ↔ ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ≤ 1 ∧ 1 ≤ ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
| 16 |
8 15
|
sylibrd |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 → ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) = 1 ) ) |