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Theorem hstel2 29206
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 29201 . . . 4 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp3bi 1098 . . 3 (𝑆 ∈ CHStates → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
32ad2antrr 762 . 2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
4 sseq1 3659 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5 fveq2 6229 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 6705 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝑦)))
76eqeq1d 2653 . . . . . . . 8 (𝑥 = 𝐴 → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝑦)) = 0))
8 oveq1 6697 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 𝑦) = (𝐴 𝑦))
98fveq2d 6233 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
105oveq1d 6705 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
119, 10eqeq12d 2666 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
127, 11anbi12d 747 . . . . . . 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 6229 . . . . . . . 8 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1514sseq2d 3666 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
16 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1716oveq2d 6706 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝐵)))
1817eqeq1d 2653 . . . . . . . 8 (𝑦 = 𝐵 → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝐵)) = 0))
19 oveq2 6698 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
2019fveq2d 6233 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
2116oveq2d 6706 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
2220, 21eqeq12d 2666 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
2318, 22anbi12d 747 . . . . . . 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 3353 . . . . 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 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))))))
2726impr 648 . . 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 750 . 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 1523  wcel 2030  wral 2941  wss 3607  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  chil 27904   + cva 27905   ·ih csp 27907  normcno 27908   C cch 27914  cort 27915   chj 27918  CHStateschst 27948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-sh 28192  df-ch 28206  df-hst 29199
This theorem is referenced by:  hstorth  29207  hstosum  29208
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