Step |
Hyp |
Ref |
Expression |
1 |
|
sto1.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
stge0 |
⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → 0 ≤ ( 𝑆 ‘ 𝐴 ) ) ) |
3 |
1 2
|
mpi |
⊢ ( 𝑆 ∈ States → 0 ≤ ( 𝑆 ‘ 𝐴 ) ) |
4 |
3
|
anim2i |
⊢ ( ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ∧ 𝑆 ∈ States ) → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑆 ‘ 𝐴 ) ) ) |
5 |
4
|
expcom |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
6 |
|
stcl |
⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) ) |
7 |
1 6
|
mpi |
⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) |
8 |
|
0re |
⊢ 0 ∈ ℝ |
9 |
|
letri3 |
⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 𝑆 ‘ 𝐴 ) = 0 ↔ ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
10 |
7 8 9
|
sylancl |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐴 ) = 0 ↔ ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑆 ‘ 𝐴 ) ) ) ) |
11 |
5 10
|
sylibrd |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 → ( 𝑆 ‘ 𝐴 ) = 0 ) ) |
12 |
|
0le0 |
⊢ 0 ≤ 0 |
13 |
|
breq1 |
⊢ ( ( 𝑆 ‘ 𝐴 ) = 0 → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ↔ 0 ≤ 0 ) ) |
14 |
12 13
|
mpbiri |
⊢ ( ( 𝑆 ‘ 𝐴 ) = 0 → ( 𝑆 ‘ 𝐴 ) ≤ 0 ) |
15 |
11 14
|
impbid1 |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐴 ) ≤ 0 ↔ ( 𝑆 ‘ 𝐴 ) = 0 ) ) |