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Theorem ishst 29378
 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 28161 . . . 4 ℋ ∈ V
2 chex 28388 . . . 4 C ∈ V
31, 2elmap 8048 . . 3 (𝑆 ∈ ( ℋ ↑𝑚 C ) ↔ 𝑆: C ⟶ ℋ)
43anbi1i 733 . 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 6347 . . . . . 6 (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
65fveq2d 6352 . . . . 5 (𝑓 = 𝑆 → (norm‘(𝑓‘ ℋ)) = (norm‘(𝑆‘ ℋ)))
76eqeq1d 2758 . . . 4 (𝑓 = 𝑆 → ((norm‘(𝑓‘ ℋ)) = 1 ↔ (norm‘(𝑆‘ ℋ)) = 1))
8 fveq1 6347 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
9 fveq1 6347 . . . . . . . . 9 (𝑓 = 𝑆 → (𝑓𝑦) = (𝑆𝑦))
108, 9oveq12d 6827 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) ·ih (𝑓𝑦)) = ((𝑆𝑥) ·ih (𝑆𝑦)))
1110eqeq1d 2758 . . . . . . 7 (𝑓 = 𝑆 → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ↔ ((𝑆𝑥) ·ih (𝑆𝑦)) = 0))
12 fveq1 6347 . . . . . . . 8 (𝑓 = 𝑆 → (𝑓‘(𝑥 𝑦)) = (𝑆‘(𝑥 𝑦)))
138, 9oveq12d 6827 . . . . . . . 8 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))
1412, 13eqeq12d 2771 . . . . . . 7 (𝑓 = 𝑆 → ((𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))
1511, 14anbi12d 749 . . . . . 6 (𝑓 = 𝑆 → ((((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦))) ↔ (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
1615imbi2d 329 . . . . 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 3123 . . . 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 749 . . 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 29376 . . 3 CHStates = {𝑓 ∈ ( ℋ ↑𝑚 C ) ∣ ((norm‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑓𝑥) ·ih (𝑓𝑦)) = 0 ∧ (𝑓‘(𝑥 𝑦)) = ((𝑓𝑥) + (𝑓𝑦)))))}
2018, 19elrab2 3503 . 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 1081 . 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 383   ∧ w3a 1072   = wceq 1628   ∈ wcel 2135  ∀wral 3046   ⊆ wss 3711  ⟶wf 6041  ‘cfv 6045  (class class class)co 6809   ↑𝑚 cmap 8019  0cc0 10124  1c1 10125   ℋchil 28081   +ℎ cva 28082   ·ih csp 28084  normℎcno 28085   Cℋ cch 28091  ⊥cort 28092   ∨ℋ chj 28095  CHStateschst 28125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-hilex 28161 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-map 8021  df-sh 28369  df-ch 28383  df-hst 29376 This theorem is referenced by:  hstcl  29381  hst1a  29382  hstel2  29383  hstrlem3a  29424
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