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Theorem hstel2 29419
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 29414 . . . 4 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp3bi 1141 . . 3 (𝑆 ∈ CHStates → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
32ad2antrr 699 . 2 (((𝑆 ∈ CHStates ∧ 𝐴C ) ∧ (𝐵C𝐴 ⊆ (⊥‘𝐵))) → ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
4 sseq1 3776 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5 fveq2 6333 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 6809 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝑦)))
76eqeq1d 2773 . . . . . . . 8 (𝑥 = 𝐴 → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝑦)) = 0))
8 fvoveq1 6817 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
95oveq1d 6809 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
108, 9eqeq12d 2786 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
117, 10anbi12d 610 . . . . . . 7 (𝑥 = 𝐴 → ((((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
124, 11imbi12d 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))))))
13 fveq2 6333 . . . . . . . 8 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1413sseq2d 3783 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15 fveq2 6333 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1615oveq2d 6810 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝐵)))
1716eqeq1d 2773 . . . . . . . 8 (𝑦 = 𝐵 → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝐵)) = 0))
18 oveq2 6802 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
1918fveq2d 6337 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
2015oveq2d 6810 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
2119, 20eqeq12d 2786 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
2217, 21anbi12d 610 . . . . . . 7 (𝑦 = 𝐵 → ((((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
2314, 22imbi12d 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))))))
2412, 23rspc2v 3473 . . . . 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 442 . . 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 687 . 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 382   = wceq 1631  wcel 2145  wral 3061  wss 3724  wf 6028  cfv 6032  (class class class)co 6794  0cc0 10139  1c1 10140  chil 28117   + cva 28118   ·ih csp 28120  normcno 28121   C cch 28127  cort 28128   chj 28131  CHStateschst 28161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-hilex 28197
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-map 8012  df-sh 28405  df-ch 28419  df-hst 29412
This theorem is referenced by:  hstorth  29420  hstosum  29421
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