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Definition df-hst 29993
 Description: Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-hst CHStates = {𝑓 ∈ ( ℋ ↑m C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
Distinct variable group:   𝑥,𝑓,𝑦

Detailed syntax breakdown of Definition df-hst
StepHypRef Expression
1 chst 28744 . 2 class CHStates
2 chba 28700 . . . . . . 7 class
3 vf . . . . . . . 8 setvar 𝑓
43cv 1537 . . . . . . 7 class 𝑓
52, 4cfv 6334 . . . . . 6 class (𝑓‘ ℋ)
6 cno 28704 . . . . . 6 class norm
75, 6cfv 6334 . . . . 5 class (norm‘(𝑓‘ ℋ))
8 c1 10527 . . . . 5 class 1
97, 8wceq 1538 . . . 4 wff (norm‘(𝑓‘ ℋ)) = 1
10 vx . . . . . . . . 9 setvar 𝑥
1110cv 1537 . . . . . . . 8 class 𝑥
12 vy . . . . . . . . . 10 setvar 𝑦
1312cv 1537 . . . . . . . . 9 class 𝑦
14 cort 28711 . . . . . . . . 9 class
1513, 14cfv 6334 . . . . . . . 8 class (⊥‘𝑦)
1611, 15wss 3908 . . . . . . 7 wff 𝑥 ⊆ (⊥‘𝑦)
1711, 4cfv 6334 . . . . . . . . . 10 class (𝑓𝑥)
1813, 4cfv 6334 . . . . . . . . . 10 class (𝑓𝑦)
19 csp 28703 . . . . . . . . . 10 class ·ih
2017, 18, 19co 7140 . . . . . . . . 9 class ((𝑓𝑥) ·ih (𝑓𝑦))
21 cc0 10526 . . . . . . . . 9 class 0
2220, 21wceq 1538 . . . . . . . 8 wff ((𝑓𝑥) ·ih (𝑓𝑦)) = 0
23 chj 28714 . . . . . . . . . . 11 class
2411, 13, 23co 7140 . . . . . . . . . 10 class (𝑥 𝑦)
2524, 4cfv 6334 . . . . . . . . 9 class (𝑓‘(𝑥 𝑦))
26 cva 28701 . . . . . . . . . 10 class +
2717, 18, 26co 7140 . . . . . . . . 9 class ((𝑓𝑥) + (𝑓𝑦))
2825, 27wceq 1538 . . . . . . . 8 wff (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))
2922, 28wa 399 . . . . . . 7 wff (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))
3016, 29wi 4 . . . . . 6 wff (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))
31 cch 28710 . . . . . 6 class C
3230, 12, 31wral 3130 . . . . 5 wff 𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))
3332, 10, 31wral 3130 . . . 4 wff 𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))
349, 33wa 399 . . 3 wff ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))
35 cmap 8393 . . . 4 class m
362, 31, 35co 7140 . . 3 class ( ℋ ↑m C )
3734, 3, 36crab 3134 . 2 class {𝑓 ∈ ( ℋ ↑m C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
381, 37wceq 1538 1 wff CHStates = {𝑓 ∈ ( ℋ ↑m C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
 Colors of variables: wff setvar class This definition is referenced by:  ishst  29995
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