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Theorem ishst 28943
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 27726 . . . 4 ℋ ∈ V
2 chex 27953 . . . 4 C ∈ V
31, 2elmap 7838 . . 3 (𝑆 ∈ ( ℋ ↑𝑚 C ) ↔ 𝑆: C ⟶ ℋ)
43anbi1i 730 . 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 6152 . . . . . 6 (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
65fveq2d 6157 . . . . 5 (𝑓 = 𝑆 → (norm‘(𝑓‘ ℋ)) = (norm‘(𝑆‘ ℋ)))
76eqeq1d 2623 . . . 4 (𝑓 = 𝑆 → ((norm‘(𝑓‘ ℋ)) = 1 ↔ (norm‘(𝑆‘ ℋ)) = 1))
8 fveq1 6152 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
9 fveq1 6152 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑦) = (𝑆𝑦))
108, 9oveq12d 6628 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) ·ih (𝑓𝑦)) = ((𝑆𝑥) ·ih (𝑆𝑦)))
1110eqeq1d 2623 . . . . . . 7 (𝑓 = 𝑆 → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ↔ ((𝑆𝑥) ·ih (𝑆𝑦)) = 0))
12 fveq1 6152 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓‘(𝑥 𝑦)) = (𝑆‘(𝑥 𝑦)))
138, 9oveq12d 6628 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))
1412, 13eqeq12d 2636 . . . . . . 7 (𝑓 = 𝑆 → ((𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
1511, 14anbi12d 746 . . . . . 6 (𝑓 = 𝑆 → ((((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
1615imbi2d 330 . . . . 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))))))
17162ralbidv 2984 . . . 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))))))
187, 17anbi12d 746 . . 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)))))))
19 df-hst 28941 . . 3 CHStates = {𝑓 ∈ ( ℋ ↑𝑚 C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
2018, 19elrab2 3352 . 2 (𝑆 ∈ CHStates ↔ (𝑆 ∈ ( ℋ ↑𝑚 C ) ∧ ((norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))))
21 3anass 1040 . 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)))))))
224, 20, 213bitr4i 292 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3559  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  0cc0 9888  1c1 9889  chil 27646   + cva 27647   ·ih csp 27649  normcno 27650   C cch 27656  cort 27657   chj 27660  CHStateschst 27690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-hilex 27726
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-sh 27934  df-ch 27948  df-hst 28941
This theorem is referenced by:  hstcl  28946  hst1a  28947  hstel2  28948  hstrlem3a  28989
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