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Theorem hstel2 28462
 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 28457 . . . 4 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp3bi 1071 . . 3 (𝑆 ∈ CHStates → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
32ad2antrr 758 . 2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
4 sseq1 3589 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 6564 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝑦)))
76eqeq1d 2612 . . . . . . . 8 (𝑥 = 𝐴 → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝑦)) = 0))
8 oveq1 6556 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 𝑦) = (𝐴 𝑦))
98fveq2d 6107 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
105oveq1d 6564 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
119, 10eqeq12d 2625 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
127, 11anbi12d 743 . . . . . . 7 (𝑥 = 𝐴 → ((((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
134, 12imbi12d 333 . . . . . 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))))))
14 fveq2 6103 . . . . . . . 8 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1514sseq2d 3596 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
16 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1716oveq2d 6565 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝐵)))
1817eqeq1d 2612 . . . . . . . 8 (𝑦 = 𝐵 → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝐵)) = 0))
19 oveq2 6557 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
2019fveq2d 6107 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
2116oveq2d 6565 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
2220, 21eqeq12d 2625 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
2318, 22anbi12d 743 . . . . . . 7 (𝑦 = 𝐵 → ((((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
2415, 23imbi12d 333 . . . . . 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))))))
2513, 24rspc2v 3293 . . . . 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))))))
2625com23 84 . . . 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))))))
2726impr 647 . . 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)))))
2827adantll 746 . 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)))))
293, 28mpd 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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   ℋchil 27160   +ℎ cva 27161   ·ih csp 27163  normℎcno 27164   Cℋ cch 27170  ⊥cort 27171   ∨ℋ chj 27174  CHStateschst 27204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-sh 27448  df-ch 27462  df-hst 28455 This theorem is referenced by:  hstorth  28463  hstosum  28464
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