Step |
Hyp |
Ref |
Expression |
1 |
|
sto1.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
stle1 |
⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ≤ 1 ) ) |
3 |
1 2
|
mpi |
⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ 𝐴 ) ≤ 1 ) |
4 |
3
|
anim1i |
⊢ ( ( 𝑆 ∈ States ∧ 1 ≤ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐴 ) ) ) |
5 |
4
|
ex |
⊢ ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆 ‘ 𝐴 ) → ( ( 𝑆 ‘ 𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
6 |
|
stcl |
⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) ) |
7 |
1 6
|
mpi |
⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) |
8 |
|
1re |
⊢ 1 ∈ ℝ |
9 |
|
letri3 |
⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 𝑆 ‘ 𝐴 ) = 1 ↔ ( ( 𝑆 ‘ 𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
10 |
7 8 9
|
sylancl |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐴 ) = 1 ↔ ( ( 𝑆 ‘ 𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
11 |
5 10
|
sylibrd |
⊢ ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆 ‘ 𝐴 ) → ( 𝑆 ‘ 𝐴 ) = 1 ) ) |
12 |
|
1le1 |
⊢ 1 ≤ 1 |
13 |
|
breq2 |
⊢ ( ( 𝑆 ‘ 𝐴 ) = 1 → ( 1 ≤ ( 𝑆 ‘ 𝐴 ) ↔ 1 ≤ 1 ) ) |
14 |
12 13
|
mpbiri |
⊢ ( ( 𝑆 ‘ 𝐴 ) = 1 → 1 ≤ ( 𝑆 ‘ 𝐴 ) ) |
15 |
11 14
|
impbid1 |
⊢ ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆 ‘ 𝐴 ) ↔ ( 𝑆 ‘ 𝐴 ) = 1 ) ) |