Step |
Hyp |
Ref |
Expression |
1 |
|
stle.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
stle.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
stle1 |
⊢ ( 𝑆 ∈ States → ( 𝐵 ∈ Cℋ → ( 𝑆 ‘ 𝐵 ) ≤ 1 ) ) |
4 |
2 3
|
mpi |
⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ 𝐵 ) ≤ 1 ) |
5 |
4
|
adantr |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( 𝑆 ‘ 𝐵 ) ≤ 1 ) |
6 |
1 2
|
stlei |
⊢ ( 𝑆 ∈ States → ( 𝐴 ⊆ 𝐵 → ( 𝑆 ‘ 𝐴 ) ≤ ( 𝑆 ‘ 𝐵 ) ) ) |
7 |
6
|
imp |
⊢ ( ( 𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑆 ‘ 𝐴 ) ≤ ( 𝑆 ‘ 𝐵 ) ) |
8 |
7
|
adantrr |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( 𝑆 ‘ 𝐴 ) ≤ ( 𝑆 ‘ 𝐵 ) ) |
9 |
|
breq1 |
⊢ ( ( 𝑆 ‘ 𝐴 ) = 1 → ( ( 𝑆 ‘ 𝐴 ) ≤ ( 𝑆 ‘ 𝐵 ) ↔ 1 ≤ ( 𝑆 ‘ 𝐵 ) ) ) |
10 |
9
|
ad2antll |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( ( 𝑆 ‘ 𝐴 ) ≤ ( 𝑆 ‘ 𝐵 ) ↔ 1 ≤ ( 𝑆 ‘ 𝐵 ) ) ) |
11 |
8 10
|
mpbid |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → 1 ≤ ( 𝑆 ‘ 𝐵 ) ) |
12 |
|
stcl |
⊢ ( 𝑆 ∈ States → ( 𝐵 ∈ Cℋ → ( 𝑆 ‘ 𝐵 ) ∈ ℝ ) ) |
13 |
2 12
|
mpi |
⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ 𝐵 ) ∈ ℝ ) |
14 |
|
1re |
⊢ 1 ∈ ℝ |
15 |
13 14
|
jctir |
⊢ ( 𝑆 ∈ States → ( ( 𝑆 ‘ 𝐵 ) ∈ ℝ ∧ 1 ∈ ℝ ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( ( 𝑆 ‘ 𝐵 ) ∈ ℝ ∧ 1 ∈ ℝ ) ) |
17 |
|
letri3 |
⊢ ( ( ( 𝑆 ‘ 𝐵 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 𝑆 ‘ 𝐵 ) = 1 ↔ ( ( 𝑆 ‘ 𝐵 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐵 ) ) ) ) |
18 |
16 17
|
syl |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( ( 𝑆 ‘ 𝐵 ) = 1 ↔ ( ( 𝑆 ‘ 𝐵 ) ≤ 1 ∧ 1 ≤ ( 𝑆 ‘ 𝐵 ) ) ) ) |
19 |
5 11 18
|
mpbir2and |
⊢ ( ( 𝑆 ∈ States ∧ ( 𝐴 ⊆ 𝐵 ∧ ( 𝑆 ‘ 𝐴 ) = 1 ) ) → ( 𝑆 ‘ 𝐵 ) = 1 ) |
20 |
19
|
exp32 |
⊢ ( 𝑆 ∈ States → ( 𝐴 ⊆ 𝐵 → ( ( 𝑆 ‘ 𝐴 ) = 1 → ( 𝑆 ‘ 𝐵 ) = 1 ) ) ) |