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Theorem hstel2 31437
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 31432 . . . 4 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp3bi 1148 . . 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 4005 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5 fveq2 6881 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
65oveq1d 7411 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝑦)))
76eqeq1d 2735 . . . . . . . 8 (𝑥 = 𝐴 → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝑦)) = 0))
8 fvoveq1 7419 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑆‘(𝑥 𝑦)) = (𝑆‘(𝐴 𝑦)))
95oveq1d 7411 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))
108, 9eqeq12d 2749 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))))
117, 10anbi12d 632 . . . . . . 7 (𝑥 = 𝐴 → ((((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)))))
124, 11imbi12d 345 . . . . . 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 6881 . . . . . . . 8 (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
1413sseq2d 4012 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15 fveq2 6881 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
1615oveq2d 7412 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) ·ih (𝑆𝑦)) = ((𝑆𝐴) ·ih (𝑆𝐵)))
1716eqeq1d 2735 . . . . . . . 8 (𝑦 = 𝐵 → (((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ↔ ((𝑆𝐴) ·ih (𝑆𝐵)) = 0))
18 oveq2 7404 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 𝑦) = (𝐴 𝐵))
1918fveq2d 6885 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑆‘(𝐴 𝑦)) = (𝑆‘(𝐴 𝐵)))
2015oveq2d 7412 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑆𝐴) + (𝑆𝑦)) = ((𝑆𝐴) + (𝑆𝐵)))
2119, 20eqeq12d 2749 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦)) ↔ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵))))
2217, 21anbi12d 632 . . . . . . 7 (𝑦 = 𝐵 → ((((𝑆𝐴) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝐴 𝑦)) = ((𝑆𝐴) + (𝑆𝑦))) ↔ (((𝑆𝐴) ·ih (𝑆𝐵)) = 0 ∧ (𝑆‘(𝐴 𝐵)) = ((𝑆𝐴) + (𝑆𝐵)))))
2314, 22imbi12d 345 . . . . . 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 3620 . . . . 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 456 . . 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 397   = wceq 1542  wcel 2107  wral 3062  wss 3946  wf 6531  cfv 6535  (class class class)co 7396  0cc0 11097  1c1 11098  chba 30137   + cva 30138   ·ih csp 30140  normcno 30141   C cch 30147  cort 30148   chj 30151  CHStateschst 30181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-hilex 30217
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-sh 30425  df-ch 30439  df-hst 31430
This theorem is referenced by:  hstorth  31438  hstosum  31439
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