Step |
Hyp |
Ref |
Expression |
1 |
|
ishst |
⊢ ( 𝑆 ∈ CHStates ↔ ( 𝑆 : Cℋ ⟶ ℋ ∧ ( normℎ ‘ ( 𝑆 ‘ ℋ ) ) = 1 ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ) ) |
2 |
1
|
simp3bi |
⊢ ( 𝑆 ∈ CHStates → ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ) |
3 |
2
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ) |
4 |
|
sseq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝐴 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ↔ ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ) ) |
8 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) ) |
9 |
5
|
oveq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) |
10 |
8 9
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) |
11 |
7 10
|
anbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ) |
12 |
4 11
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ) ) |
13 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( ⊥ ‘ 𝑦 ) = ( ⊥ ‘ 𝐵 ) ) |
14 |
13
|
sseq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝐵 ) ) |
16 |
15
|
oveq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) ) |
17 |
16
|
eqeq1d |
⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ↔ ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ) ) |
18 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ∨ℋ 𝑦 ) = ( 𝐴 ∨ℋ 𝐵 ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ) |
20 |
15
|
oveq2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) |
21 |
19 20
|
eqeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) |
22 |
17 21
|
anbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
23 |
14 22
|
imbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) ) |
24 |
12 23
|
rspc2v |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) ) |
25 |
24
|
com23 |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) ) |
26 |
25
|
impr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
27 |
26
|
adantll |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆 ‘ 𝑥 ) ·ih ( 𝑆 ‘ 𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑆 ‘ 𝑦 ) ) ) ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) ) |
28 |
3 27
|
mpd |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) |