Step |
Hyp |
Ref |
Expression |
1 |
|
isst |
⊢ ( 𝑆 ∈ States ↔ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) ) |
2 |
1
|
simp3bi |
⊢ ( 𝑆 ∈ States → ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) |
3 |
|
sseq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ) ) |
4 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝐴 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) |
7 |
4 6
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) |
8 |
3 7
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( ⊥ ‘ 𝑦 ) = ( ⊥ ‘ 𝐵 ) ) |
10 |
9
|
sseq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ∨ℋ 𝑦 ) = ( 𝐴 ∨ℋ 𝐵 ) ) |
12 |
11
|
fveq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ) |
13 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝐵 ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) |
15 |
12 14
|
eqeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) |
16 |
10 15
|
imbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
17 |
8 16
|
rspc2v |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
18 |
2 17
|
syl5com |
⊢ ( 𝑆 ∈ States → ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
19 |
18
|
impd |
⊢ ( 𝑆 ∈ States → ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) |