Metamath Proof Explorer

Theorem hstel2

Description: Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hstel2 ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) )

Proof

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 ) ) ) )