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Theorem ishst 32242
Description: Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ishst (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem ishst
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 31027 . . . 4 ℋ ∈ V
2 chex 31254 . . . 4 C ∈ V
31, 2elmap 8909 . . 3 (𝑆 ∈ ( ℋ ↑m C ) ↔ 𝑆: C ⟶ ℋ)
43anbi1i 624 . 2 ((𝑆 ∈ ( ℋ ↑m C ) ∧ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))) ↔ (𝑆: C ⟶ ℋ ∧ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))))
5 fveq1 6905 . . . . 5 (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
65fveqeq2d 6914 . . . 4 (𝑓 = 𝑆 → ((norm‘(𝑓‘ ℋ)) = 1 ↔ (norm‘(𝑆‘ ℋ)) = 1))
7 fveq1 6905 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
8 fveq1 6905 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑦) = (𝑆𝑦))
97, 8oveq12d 7448 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) ·ih (𝑓𝑦)) = ((𝑆𝑥) ·ih (𝑆𝑦)))
109eqeq1d 2736 . . . . . . 7 (𝑓 = 𝑆 → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ↔ ((𝑆𝑥) ·ih (𝑆𝑦)) = 0))
11 fveq1 6905 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓‘(𝑥 𝑦)) = (𝑆‘(𝑥 𝑦)))
127, 8oveq12d 7448 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))
1311, 12eqeq12d 2750 . . . . . . 7 (𝑓 = 𝑆 → ((𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
1410, 13anbi12d 632 . . . . . 6 (𝑓 = 𝑆 → ((((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
1514imbi2d 340 . . . . 5 (𝑓 = 𝑆 → ((𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))) ↔ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
16152ralbidv 3218 . . . 4 (𝑓 = 𝑆 → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
176, 16anbi12d 632 . . 3 (𝑓 = 𝑆 → (((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))))) ↔ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))))
18 df-hst 32240 . . 3 CHStates = {𝑓 ∈ ( ℋ ↑m C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
1917, 18elrab2 3697 . 2 (𝑆 ∈ CHStates ↔ (𝑆 ∈ ( ℋ ↑m C ) ∧ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))))
20 3anass 1094 . 2 ((𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))) ↔ (𝑆: C ⟶ ℋ ∧ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))))
214, 19, 203bitr4i 303 1 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wss 3962  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  0cc0 11152  1c1 11153  chba 30947   + cva 30948   ·ih csp 30950  normcno 30951   C cch 30957  cort 30958   chj 30961  CHStateschst 30991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-sh 31235  df-ch 31249  df-hst 32240
This theorem is referenced by:  hstcl  32245  hst1a  32246  hstel2  32247  hstrlem3a  32288
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