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Theorem hstel2 30012
 Description: Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hstel2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))

Proof of Theorem hstel2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishst 30007 . . . 4 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp3bi 1144 . . 3 (𝑆 ∈ CHStates → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
32ad2antrr 725 . 2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
4 sseq1 3940 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5 fveq2 6646 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 7151 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝑦)))
76eqeq1d 2800 . . . . . . . 8 (𝑥 = 𝐴 → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝑦)) = 0))
8 fvoveq1 7159 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
95oveq1d 7151 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
108, 9eqeq12d 2814 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
117, 10anbi12d 633 . . . . . . 7 (𝑥 = 𝐴 → ((((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
124, 11imbi12d 348 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))))
13 fveq2 6646 . . . . . . . 8 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1413sseq2d 3947 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15 fveq2 6646 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1615oveq2d 7152 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝐵)))
1716eqeq1d 2800 . . . . . . . 8 (𝑦 = 𝐵 → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝐵)) = 0))
18 oveq2 7144 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
1918fveq2d 6650 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
2015oveq2d 7152 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
2119, 20eqeq12d 2814 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
2217, 21anbi12d 633 . . . . . . 7 (𝑦 = 𝐵 → ((((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
2314, 22imbi12d 348 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))))
2412, 23rspc2v 3581 . . . . 5 ((𝐴C𝐵C ) → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) → (𝐴 ⊆ (⊥‘𝐵) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))))
2524com23 86 . . . 4 ((𝐴C𝐵C ) → (𝐴 ⊆ (⊥‘𝐵) → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))))
2625impr 458 . . 3 ((𝐴C ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
2726adantll 713 . 2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
283, 27mpd 15 1 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  0cc0 10529  1c1 10530   ℋchba 28712   +ℎ cva 28713   ·ih csp 28715  normℎcno 28716   Cℋ cch 28722  ⊥cort 28723   ∨ℋ chj 28726  CHStateschst 28756 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-hilex 28792 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-map 8394  df-sh 29000  df-ch 29014  df-hst 30005 This theorem is referenced by:  hstorth  30013  hstosum  30014
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